Bifurcation analysis of reaction–diffusion Schnakenberg model

Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling Schnakenberg chemical reaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns.     

Atom diffusion in furnaces — models and measurements

Experimental as well as the theoretical approach to estimate diffusion coefficients for several analyte elements with different behavior in graphite furnaces, lead, gold, indium and chromium, were investigated. ‘Close’ graphite furnaces of two designs differing in the size of end apertures and the diameter of the injection port were used. The furnaces were fast heated at rates of approximately 10 000 K s⁻¹. The peak absorbance of all studied analytes was independent of geometry, suggesting that the separation of atomization and removal was attained. Residence times of the analytes in the two different furnaces were determined from absorbance tail shapes. In the case of gold, the influence of temperature in the range between 1800 and 2200 K on the residence time in both furnaces was also found. The residence times measured in the two different furnaces under otherwise identical conditions, made possible to select the accurate model of diffusional removal from several possible models. The knowledge of the accurate model allowed the estimate of experimental diffusion coefficients. They were thus compared with those semiempirically calculated from kinetic theory of gases, extended to allow for the intermolecular forces. The accuracy of these calculations is limited since the input data (critical temperatures, boiling temperature or melting temperature, molal volumes at the critical, boiling and melting points, metallic crystallographic radii and dissociation constants of metal dimers) are not known with adequate accuracy. The comparison of ‘theoretical’ and ‘experimental’ values of diffusion coefficients makes possible to assess value of using individual sources of input data for the semiempirical calculations.     

Traveling wave patterns in nonlinear reaction–diffusion equations

In this paper we study a reaction–diffusion model equation with general nonlinear diffusion and arbitrary kinetic orders in the reaction terms, which appears in the applied biochemical modeling. We carry both analytical and numerical studies of the model equation to show the existence of monotone and oscillatory waves. Our numerical computations are illustrated for a particular case of the equation by using different methods which lead to accurate wave profiles and confirm the analytical results.     

Hybrid stochastic simulations of intracellular reaction–diffusion systems

With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction–diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a hybrid stochastic approach is introduced for simulating reaction–diffusion systems. We developed an adaptive partitioning strategy in which processes with high frequency are simulated with deterministic rate-based equations, and those with low frequency using the exact stochastic algorithm of Gillespie. Therefore the stochastic behavior of cellular pathways is preserved while being able to apply it to large populations of molecules. We describe our method and demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors.     

Are steady state diffusion layer thicknesses potential independent?

The required conditions are considered for the potential-independence of steady state diffusion layer thickness in an electrochemical experiment. Such conditions are illustrated by numerical solution of the mass transport equations for an electroactive species at the microdisc and tubular flow electrodes. Published in Russian in Elektrokhimiya, 2008, Vol. 44, No. 1, pp. 134–141. The text was submitted by the authors in English.     

Reactive diffusion in W–Mo–Si thin films

This study reports the phase formation in the ternary thin films system Mo–W–Si. The metallic films were deposited onto Si (100) substrate by sputtering. Two kinds of samples were prepared, either by sequential deposition or by co-deposition. The phase formation was investigated by In situ X-ray diffraction measurements from 300 to 900 °C. The influence of the sample preparation, namely sequential deposition and co-deposition, on the mechanism of phase formation has been evidenced.     

“Up-hill” diffusion and hydrogen-hydrogen interaction in palladium-platinum alloys

The reported anomalous “up-hill” diffusion of hydrogen in PdPt and PdAg alloys is explained in terms of attractive, long-range, non-local, elastic interactions between dissolved hydrogen atoms. A statistical model based on the continuum elasticity theory is presented, and model calculations for the Pd₈₁Pt₁₉ system are shown to give results that are in semi-quantitative agreement with experimental data.     

A digital simulation study of two-dimensional convection diffusion for channel electrode

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Can "microwave effects" be explained by enhanced diffusion?

The "microwave effect" or non-thermal effects due to microwaves have been the subject of intense debate. This paper explores the following hypothesis: if the transport of an active species is a rate limiting step in a reaction, and if microwaves enhance the diffusion of that species, then the overall reaction rate would change under microwave heating compared with conventional heating. If the hypothesis is correct then it should be possible to pick those reactions that would speed up, slow down or stay the same, under microwave irradiation. One consequence of the hypothesis is that the equilibrium states (end point of the reactions) remain unchanged by microwave irradiation. The measurements and theory presented here strongly suggest that this hypothesis is correct.     

Effects of different sized concentration differences across free diffusion boundaries and comparison of Gouy and Rrayleigh diffusion measurements using NaCl−KCl−H2O

Diffusion was systematically studied in the ternary system NaCl (0.5M)–KCl (0.5M)–H₂O at 25°C. There were four purposes. First, current methods to extractD _ij from Gouy and Rayleigh interferometry data depend on treating theD _ij as effectively constant. If concentration differences C _i across the boundary are large, this may not be true. To explore this issue, four sets of experiments were performed. Each set had four experiments with approximately the same total number of fringesJ. Each set also had the same corresponding C ₁/C ₂ ratios as the other three, but the C _i were adjusted such that the four sets hadJ30, 60, 90, and 120, respectively. Noclear dependence of theD _ij on C _i was found within their realistic errors. Second, the Gosting diffusiometer can yield both Rayleigh and Gouy fringe patterns during the same experiment. Therefore, theD _ij from both methods were compared, and agree well. Third, a new method for analyzing Gouy fringe positions (Miller; program TNY) can be compared to the classical one (Fujita-Gosting; program RFG). TheD _ij from both analyses agree well. Fourth, we compared results from the Gosting diffusiometer with those at the same composition from other diffusiometers: one data set by O[Donnell and Gosting from an older Gouy apparatus, and three Rayleigh sets at differentJ values from our older Model H diffusiometer. Results from older diffusiometers were more scattered, but correspondingD _ij agree within realistic errors. As reported previously, realistic errors are approximately four times the statistical errors obtained by least squares. Recommended Rayleigh and GouyD _ij are presented for this composition.     

A second-order scheme for the “Brusselator” reaction–diffusion system

A second-order method is developed for the numerical solution of the initial-value problems , , and , , , in which the functions and , where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point ( , V* = A/B) provided . The approach adopted is extended to solve a class of non-linear reaction–diffusion equations in two-space dimensions known as the “Brusselator” system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs). This revised version was published online in July 2006 with corrections to the Cover Date.     

Improved diffusion Monte Carlo propagators for bosonic systems using Itô calculus

The construction of importance sampled diffusion Monte Carlo (DMC) schemes accurate to second order in the time step is discussed. A central aspect in obtaining efficient second order schemes is the numerical solution of the stochastic differential equation (SDE) associated with the Fokker-Plank equation responsible for the importance sampling procedure. In this work, stochastic predictor-corrector schemes solving the SDE and consistent with It? calculus are used in DMC simulations of helium clusters. These schemes are numerically compared with alternative algorithms obtained by splitting the Fokker-Plank operator, an approach that we analyze using the analytical tools provided by Ito; calculus. The numerical results show that predictor-corrector methods are indeed accurate to second order in the time step and that they present a smaller time step bias and a better efficiency than second order split-operator derived schemes when computing ensemble averages for bosonic systems. The possible extension of the predictor-corrector methods to higher orders is also discussed.     

On the diffusion of free carriers in β-rhombohedral boron

To determine the diffusion of untrapped carriers in β-rhombohedral boron, we constructed a feedback pico-ammeter based on pulse integration technique. This enabled measuring deviations from the bias in a 10⁹ Ω sample in the order of 1 nA with 0.7 ms time resolution. For the first time, we obtained the drift velocity of optically generated untrapped electron-hole pairs 106(20) cm s⁻¹ yielding for the band-determined diffusion coefficient and for the carrier mobility . Fitting Fick's second law to the measured trap-determined dispersion of carriers yields the ambipolar diffusion coefficient D^*=0.043(14) and 0.28(10) cm² s⁻¹ at 260 and 340 K, respectively. The thermal activation energy of 0.18 eV agrees with the well-known trapping levels in β-rhombohedral boron.     

Maximum entropy-based variational multiscale element-free Galerkin methods for scalar advection–diffusion problems

Journal of Thermal Analysis and Calorimetry - This paper presents a new parameter-free stability term via Galerkin mesh-free scheme to solve the multiscale behavior of steady-state...     

Differential diffusion quantum Monte Carlo method: determination of potential energy surfaces of molecules

A differential approach for self-optimizing diffusion Monte Carlo calculation was proposed in this paper, which is a new algorithm combining three techniques such as optimizing, diffusion and correlation sampling. This method can be used to directly compute the energy differential between two systems in the diffusion process, making the statistical error of calculation be reduced to order of 10-5 hartree, and recover about more than 80% of the correlation energy. We employed this approach to set up a potential energy surface of a molecule, used a "rigid move" model, and utilized Jacobi transformation to make energy calculation for two configurations of a molecule having good positive correlation. So, an accurate energy differential could be obtained, and the potential energy surface with good quality can be depicted. In calculation, a technique called "post-equilibrium remaining sample was set up firstly, which can save about 50% of computation expense. This novel algorithm was used to study the potenti     

Sedimentation–diffusion profiles and layered sedimentation of charged colloids at low ionic strength

We report the formation of strongly inflated sedimentation–diffusion concentration profiles for charged monodisperse colloidal spheres in absolute ethanol. Various additional experiments, such as light scattering, confirm that the very dilute supernatants, left behind by the majority of settling colloids, contain spheres that repel each other at distances of micrometers. We attribute these unusual profiles to a significant counter-ion contribution to the osmotic pressure and to the Debye screening length. An approximate osmotic equation-of-state at the level of the second virial coefficient for dispersions at very low ionic strength indeed implies an algebraic long-distance decay of sedimentation–diffusion profiles, together with significant lowering of the effective colloid mass by an entropic lift due to counter-ions. We have also observed that sedimenting dispersions sometimes demix into two layers, which are both disordered fluids. Since the colloids are clearly repulsive on the DLVO pair level, this layering possibly manifests a phase transition driven by many-body attractions.     

Adsorption and diffusion of propane and propylene in Ag+-impregnated MCM-41

Equilibrium data and diffusion characteristics of propane and propylene were determined on mesoporous adsorbents modified with an organic molecule (APTES) and/or impregnated with AgNO₃, in order to obtain a separation by adsorption via π-complexation. Adsorption capacities were determined by a gravimetric technique, while diffusion characteristics were evaluated by the ZLC technique. The equilibrium isotherms data showed that the modification with an organic molecule will weaken the π-interaction between Ag⁺ and double C=C bond. On the other hand slightly higher adsorption capacities for propylene (about 1.5 mol/kg) were obtained for the sample prepared by a direct impregnation with larger amounts of AgNO₃ (M4 sample). Diffusion runs confirmed that the propane desorption rate on M4 sample was much higher compared to propylene. This evidence leads to a potential application of that adsorbent material for a kinetic separation.     

Interstitialcy diffusion of oxygen in tetragonal La2CoO(4+δ)

We report on the mechanism and energy barrier for oxygen diffusion in tetragonal La(2)CoO(4+δ). The first principles-based calculations in the Density Functional Theory (DFT) formalism were performed to precisely describe the dominant migration paths for the interstitial oxygen atom in La(2)CoO(4+δ). Atomistic simulations using molecular dynamics (MD) were performed to quantify the temperature dependent collective diffusivity, and to enable a comparison of the diffusion barriers found from the force field-based simulations to those obtained from the first principles-based calculations. Both techniques consistently predict that oxygen migrates dominantly via an interstitialcy mechanism. The single interstitialcy migration path involves the removal of an apical lattice oxygen atom out from the LaO-plane and placing it into the nearest available interstitial site, whilst the original interstitial replaces the displaced apical oxygen on the LaO-plane. The facile migration of the interstitial oxygen in this path is enabled by the cooperative titling-untilting of the CoO(6) octahedron. DFT calculations indicate that this process has an activation energy significantly lower than that of the direct interstitial site exchange mechanism. For 800-1000 K, the MD diffusivities are consistent with the available experimental data within one order of magnitude. The DFT- and the MD-predictions suggest that the diffusion barrier for the interstitialcy mechanism is within 0.31-0.80 eV. The identified migration path, activation energies and diffusivities, and the associated uncertainties are discussed in the context of the previous experimental and theoretical results from the related Ruddlesden-Popper structures.     

99TcO4 − diffusion and sorption in compacted GMZ bentonite studied by capillary method

Sorption and diffusion of ⁹⁹TcO₄ ^? in compacted bentonite from Gaomiaozi county (Inner Mongolia, China) (denoted as GMZ bentonite) was studied by capillary method. The distribution coefficient (K _d) and diffusion coefficient (D _a) were calculated from diffusion curves of ⁹⁹TcO₄ ^? in compacted GMZ bentonite. The D _a and K _d values decreased with the increasing of the bulk dry density of the compacted bentonite. With increasing pH values, the D _a values increased whereas the K _d values decreased, which is attributed to the ion exchange and electrostatic repulsion between the negative surface charged bentonite and negative charged ⁹⁹TcO₄ ^? ions at high pH values. The diffusion and relative concentration distributions of ⁹⁹TcO₄ ^? in compacted bentonite after long aging time were simulated from the distribution and diffusion coefficients. Effect of aging time on the diffusion of ⁹⁹TcO₄ ^? in compacted bentonite was also calculated using the theoretical equation. The results indicated that ⁹⁹TcO₄ ^? was an easily mobile radionuclide in environment and the release of ⁹⁹TcO₄ ^? from the radioactive waste repository was dangerous to the environmental pollution.     

The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry

In this paper, we have applied an accurate and efficient wavelet scheme (due to Legendre polynomial) to find the numerical solutions for a set of coupled reaction–diffusion equations. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of non linear terms appearing in the governing differential equations. The highest derivative in the differential equation is expanded into wavelet series, this approximation is then integrated while the boundary conditions are applied by using integration constants. With the help of operational matrices, the nonlinear reaction–diffusion equations are converted into a system of algebraic equations. Finally, some numerical examples to demonstrate the validity and applicability of the method have been furnished. The use of Legendre wavelets is found to be accurate, efficient, simple, and computationally attractive. This wavelet method can be used for obtaining quick solution in many chemical Engineering problems.