Two-Cell coupled cubic autocatalator: the effect of the uncatalysed reaction

A two-cell coupled chemical oscillator based on the cubic autocatalator: P→AA + 2B→ 3BB → C is considered with the inclusion of the extra uncatalysed reaction A→ B. Two cases are considered. In each the cells are assumed to be identical, with the coupling between them taking place by a linear exchange of either reactant A or autocatalyst B. In the former case parameter ranges are found over which it is possible to have one, three or five stationary states; these are discussed in detail, as well as the points of Hopf bifurcation on the various branches of solution. In the latter case, it is shown that the only stationary state possible is the same as for the uncoupled system with Hopf bifurcations also at the same parameter values as for the uncoupled system for all values of the non-dimensional constant describing the coupling between the cells.

Asymptotic solutions are derived for large values of the coupling constants in both cases. With coupling through the autocatalyst the system is shown to reduce to the uncoupled system, whilst for coupling through A the former asymmetric solutions are still found, with Hopf bifurcations on these solution branches     

Differential-flow-induced instability in a cubic autocatalator system

The formation of spatio-temporal stable patterns is considered for a reaction-diffusion-convection system based upon the cubic autocatalator, A + 2B 3B, B C, with the reactant A being replenished by the slow decay of some precursor P via the simple step P A. The reaction is considered in a differential-flow reactor in the form of a ring. It is assumed that the reactant A is immobilised within the reactor and the autocatalyst B is allowed to flow through the reactor with a constant velocity as well as being able to diffuse. The linear stability of the spatially uniform steady state (a, b) = (µ^–1, µ), where a and b are the dimensionless concentrations of the reactant A and autocatalyst B, and µ is a parameter reflecting the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, flow-generated patterns is that the flow parameter > _c(µ, ) where _c(µ,) is a (strictly positive) critical value of and is the dimensionless diffusion coefficient of the species B and also reflects the size of the system. Values of _c at which these bifurcations occur are derived in terms of µ and . Further information about the nature of the bifurcating branches (close to their bifurcation points) is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical Hopf bifurcations are possible. The bifurcating branches are then followed numerically by means of a path-following method, with the parameter as a bifurcation parameter, for representatives values of µ and . It is found that multiple stable patterns can exist and that it is also possible that any of these can lose stability through secondary Hopf bifurcations. This typically gives rise to spatio-temporal quasiperiodic transients through which the system is ultimately attracted to one of the remaining available stable patterns.     

Existence and uniqueness of periodic orbits in the cubic autocatalator

Periodic orbits are sought in a mathematical model of a simple prototype chemical reaction involving essentially only two reacting species. Physically, these periodic orbits correspond to time-periodic oscillations in the concentrations of the two chemicals. Using the results for the existence and uniqueness of periodic orbits for Lienard systems, necessary and sufficient conditions are obtained for the existence of exactly one periodic orbit and of no periodic orbits. The results apply to a closed system where the quadratic autocatalytic reaction and decay step are present but the uncatalysed reaction is not and where there is only one physically relevant equilibrium solution     

The cubic autocatalator: The influence of degenerate singularities in a closed system

The appearance of apparently chaotic behaviour in this two-dimensional system is examined from an analytical point of view. The original two-parameter model exhibiting numerical solutions resembling chaos is unfolded to a three-parameter model. This enlarged model is shown to have a codimension-two degenerate Hopf bifurcation the unfolding of which contains phase portraits showing three concentric limit cycles. In some regions these limit cycles are so close to each other that numerical integration causes transitions across the unstable limit cycle, giving the appearance of chaotic behaviour. The region in parameter space of the ‘chaotic’ behaviour agrees well with the degenerate behaviour of the enlarged model.     

The cubic autocatalator: The influence of the quadratic autocatalytic and the uncatalysed reactions

The proptotype chemical reaction scheme, the cubic autocatalator, A + 2B 3B; B C is taken in a closed system, with A formed from the precursor P by the simple step P A. The pooled chemical approximation is invoked whereby the concentration of P can be assumed to remain constant throughout. The effects of allowing the quadratic autocatalytic reaction A + B 2B and the uncatalysed reaction A B in the scheme are considered in detail. The full scheme is described by the non-dimensional parameters µ (measuring the reaction rate of the initiation step) ands andr (measuring the reaction rates of the quadratic autocatalytic and the uncatalysed steps respectively). It is shown, provided only thatr ors (or both) are non-zero, no matter how small, the solution remains bounded for all (positive) values of µ, whereas withr =s = 0 the solution is bounded only for µ > µ₀ (µ₀ = 0.900 32). It is shown that withr = 0 ands 0 the governing equations have a Hopf bifurcation at µ = 1 –s producing a stable limit cycle which exists for all µ in 0 < µ <1 –s. The behaviour of these limit cycles as µ 0 is also discussed.     

A model for whisker growth limited by a heterogeneous chemical reaction

A model has been proposed for whisker growth controlled by a heterogeneous chemical reaction on the surface of the liquid phase. The model considers a “feed” zone inside the boundary layer in the vapor phase, where the gas flow in the reactor has a negligible velocity. Since the feed zone is considerably larger than the crystal, the model possesses spherical symmetry. In the immediate vicinity of the surface of the liquid phase, there is a gas layer with constant reactant concentrations. The growth rate is determined by the reactant concentrations in this layer, which are controlled by the balance between the diffusion and chemical processes involved. Expressions are obtained for the concentrations of the components at the surface of the liquid phase and for the crystal growth rate as a function of crystal radius. The etching of a crystal through the liquid phase is considered.     

Numerical study of the motion behaviour of three-dimensional cubic particle in a thin drum

The motion of three-dimensional cubic particles in a thin rotating drum is simulated by the SIPHPM method. The drums with frictional or smooth front and rear walls, and the particles of cubic and spherical shapes, and different particle numbers are considered to study the effect of cubic particle shape, end-wall frictions and filling levels. Different flow patterns of cubic particles are observed, which are significantly dominated by the friction from the end-walls. The probability density function of velocity components, the flatness factors are used to analyze the motion behaviour of cubic particle. The Froude number, ensemble mean and time averaged particle velocities are also analyzed. A primary and secondary mode of driving from the end-wall frictions are indicated and the mechanisms on the influences of wall friction, particle shape and filling levels are fully explained.     

Convex forms of the cubic model

In the paper a generalised model is proposed which encompasses a number of known models of the mechanical material behaviour: the normal stress hypothesis, the maximum strain hypothesis, rotationally symmetric potentials (v. MISES, BELTRAMI, cone, paraboloid- and hyperboloid potentials) and the model of SAYIR. The model gives rise to some new solutions which extend the classic hypotheses. The model is build up of the three invariants of the stress tensor and has five free parameters. Parameters are restricted to yield a convex form. Some convex forms of the model, which contains the stresses to the power three or two, are constructed in the paper.     

Effect of reaction pressure on the nucleation behaviour of diamond synthesized by hot-filament chemical vapour deposition

Synthetic diamond particles were deposited on a Si (1 0 0) substrate using a hot-filament chemical-vapour-deposition method in order to study the effect of the reaction pressure on the nucleation behaviour. The reaction pressure was controlled, as an experimental variable, from 2 to 50 torr under the following conditions: a filament temperature of 2200 °C, a substrate temperature of 850 °C, a total flow rate of 200 s.c.c.m. and a methane concentration of 0.8 vol%. Diamond deposits on the Si wafer were characterized by micro-Raman spectroscopy, scanning electron microscopy (SEM) and optical microscopy.The maximum nucleation density of diamond particles on the unscratched Si substrate is shown at the reaction pressure of 5 torr. These phenomena can be explained by the competition effect between -SiC formation, which increases the diamond nucleation density, and atomic-hydrogen etching which decreases the nucleation sites.A new fabrication method for a high-quality diamond film without any surface pretreatments is introduced using a combination process between diamond nucleation at low pressure (5 torr) and growth at high pressure (30 torr).     

On the behaviour of body-centred cubic metals to one-dimensional shock loading

The response of metallic materials to shock loading, like all loading regimes, is controlled largely by factors operating at the microscopic or atomic levels. Over the past few years, face-centred cubic (fcc) metals have received a level of attention where the role of features such as stacking fault energy and precipitation hardening have been investigated. We now turn our attention to body-centred cubic (bcc) metals. In the past, only tantalum, tungsten, and their alloys have received significant attention at high strain-rate conditions due to their use by the ordnance community. In particular, this investigation examines the shear strength of these materials at shock loading conditions. Previous results on tantalum, tungsten, and a tungsten heavy alloy are reviewed, and more recent experiments on niobium, molybdenum, and Ta–2.5 wt% W presented. Results are discussed in terms of known deformation mechanisms and variations of Peierl’s stress.     

Mass-transfer mechanism in a vibrationally fluidized chemical reaction dispersion

An elementary analysis is applied to the mass transport mechanism in a solid-state chemical process occurring in a vibrating finely divided mixture. The analysis is confirmed by experiment.Translated from Inzhenerno-fizicheskii Zhurnal, No. 2, pp. 209–217, February, 1991.     

Mechanisms of prion disease progression: a chemical reaction network approach

Fatal neurodegenerative diseases such as bovine spongiform encephalopathy in cattle, scrapie in sheep and Creutzfeldt-Jakob disease in humans are caused by prions. Prion is a protein encoded by a normal cellular gene. The cellular form of the prion, namely PrP(C), is benign but can be converted into a disease-causing form (named scrapie), PrP(Sc), by a conformational change from -helix to -sheets. Prions replicate by this conformational change; that is, PrP(Sc) interacts with PrP(C) producing a new molecule of PrP(Sc). This kind of replication is modelled in this contribution as an autocatalytic process. The kinetic model accounts for two of the three epidemiological manifestations: sporadic and infectious. By assuming irreversibility of the PrP(Sc) replication and describing a first-order reaction for the degradation of cellular tissue, the authors explore dynamical scenarios for prion progression, such as oscillations and conditions for multiplicity of equilibria. Feinberg's chemical reaction network theory is exploited to identify multiple steady states and their associate kinetic constants.     

Monotonicity in chemical reaction systems

This article discusses the question of when the dynamical systems arising from chemical reaction networks are monotone, preserving an order induced by some proper cone. The reaction systems studied are defined by the reaction network structure while the kinetics is only constrained very weakly. Necessary and sufficient conditions on cones preserved by these systems are presented. Linear coordinate changes which make a given reaction system cooperative are characterized. Also discussed is when a reaction system restricted to an invariant subspace is cone preserving, even when the system fails to be cone preserving on the whole of phase space. Many of the proofs allow explicit construction of preserved cones. Numerous examples of chemical reaction systems are presented to illustrate the results.     

Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme

The formation of stable patterns is considered in a reaction-diffusion system based on the cubic autocatalator, A+2B 3B, B C, with the reaction taking place within a closed region, the reactant A being replenished by the slow decay of precursor P via the reaction P A. The linear stability of the spatially uniform Steady state % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaabmGabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyyp% a0ZaaeWaceaacqaH8oqBdaahaaWcbeqaaiabgkHiTiaaigdaaaGcca% GGSaGaeqiVd0gacaGLOaGaayzkaaaaaa!48C3!\[\left( {a,b} \right) = \left( {\mu ^{ - 1} ,\mu } \right)\], where a and b are the dimensionless concentrations of reactant A and autocatalyst B and is a parameter representing the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, solutions (patterns) is that the parameter % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadseacqGH8aapcaaIZaGaeyOeI0IaaGOmamaakaaabaGaaGOm% aaWcbeaaaaa!4139!\[D < 3 - 2\sqrt 2 \] where D=D _b/D_a (D _a and D _b are the diffusion coefficients of chemical species A and B respectively). The values of at which these bifurcations occur are derived in terms of D and (a parameter reflecting the size of the system). Further information about the nature of the spatially non-uniform solutions close to their bifurcation points is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical bifurcations are possible. The bifurcation branches are then followed numerically using a path-following method, with as the bifurcation parameter, for representative values of D and . It is found that the stable patterns can lose stability through supercritical Hopf bifurcations and these stable, temporally periodic, spatially non-uniform solutions are also followed numerically.     

Oscillatory behaviour in a three-component plankton population model

We examine the qualitative behaviour of an NPZ (nutrient-phyto- plankton-zooplankton) model for parameter ranges consistent with values used in the literature. The wide range of values partly reflects variations of conditions in different environtments for the plankton, but in many cases is a measure of the difficulties in making observations and consequent uncertainties. We pay particular attention to the bifurcational behaviour of the system, and to the regions of parameter space for which oscillatory behaviour is possible; such oscillatory behaviour has recently been found in both observational data and in more complex ecosystem models. In some regions of parameter space, we also find that multiple attractors occur. Finally, we examine in more detail the behaviour for a range of values of nutrient input.     

Microwave-enhanced chemical vapor infiltration: a sharp interface model

A simple one-dimensional model for describing CVI into a fibrous preform is presented and analyzed. The derivation requires that the thermal and electrical properties of the preform and substrate are disparate. Mathematically the model takes the form of a moving-boundary problem, where the boundary is the interface between the preform and the substrate. Effects of heat loss and radiation at the boundaries and the nonlinear dependence of dielectric parameters on temperature are all taken into account, in addition to the spatial dependence on gas concentration. Processing times for different power levels and inlet concentrations for different initial substrate thicknesses are obtained, and the implications to processing strategies are discussed.     

Micromixing models for calculating the average rate of a chemical reaction in a turbulent flow

A model of mixing with a chemical reaction in a uniform, isotropic turbulent flow has been developed on the basis of the statistical-moments method for calculating the hydrodynamic characteristics of a flow and the probability density function of its mixture coefficient for the purpose of determining the correlations responsible for the chemical reaction in the flow. The results of calculations of the average rate of a chemical reaction in a flow by the multizone model and models of micromixing (the mean-square approximation and Langevin models) used for construction of equations for the probability density function were compared.     

Occurrence of free convection in a plane layer in the presence of a chemical transport reaction

We consider the conditions for the occurrence of natural convection when a volatile compound is transported through a gaseous phase as a result of a chemical reaction with a solid substance. We determine the variation of the critical Rayleigh number for the principal level of instability as a function of the parameters of the process.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 38, No. 1, pp. 99–106, January, 1980.     

Water as a reaction medium for clean chemical processes

Solvent usage is often an integral part of manufacturing process, whether it is chemical or another industrial sector. Thus, this unavoidable choice of a specific solvent for a desired manufacturing process can have profound economical, environmental, and societal implications. Some of the impacts are long lasting especially from an environmental perspective, which has been well documented in the scientific literature. The pressing need to develop alternative solvents for manufacturing processes originates, in part, from these implications and constitutes an essential strategy under an emerging field of green chemistry. Whereas there have been excellent advances in developing several alternative clean solvents, it is unlikely that the one solvent will be a panacea for various chemical protocols. This article provides some examples of using water as an alternative solvent for chemical reactions with wide-ranging possibilities that include direct use of water soluble renewable materials, C–C bond forming reactions using organometallic reagents, and exploiting the use of alternate energy sources such as solar, microwave and ultrasound in accelerating chemical syntheses.An erratum to this article can be found at