Initial Distribution Of Reactants Research Articles

Simulation of reactive mixing behaviors inside micro-droplets by a lattice Boltzmann method

The Villermaux/Dushman competing parallel reactions are modeled into a multiphase/multicomponent lattice Boltzmann method (LBM) to theoretically investigate the interplay between mixing and reactions in droplet-based microfluidics. To quantify the mixing and reaction efficiency inside the droplets, the standard deviation σ representing the pure mixing performance without reaction and the segregation index Xs reflecting mixing index with simultaneous chemical reactions are calculated based on the simulation results. The results show that the species transport can be evidently intensified by matching the flow recirculation and the appropriate initial distribution of reactants. We further demonstrate that the reaction selectivity could be improved by tuning droplet flow conditions or adding internal baffles to arbitrarily adjust the position of recirculation inside droplet.

The general theory of multistage geminate reactions of isolated pairs of reactants. III. Two-stage reversible dissociation in geminate reaction A + A ↔ C ↔ B + B.

Specific two-stage reversible reaction A + A ↔ C ↔ B + B of the decay of species C reactants by two independent transition channels is considered on the basis of the general theory of multistage reactions of isolated pairs of reactants. It is assumed that at the initial instant of time, the reacting system contains only reactants C. The employed general approach has made it possible to consider, in the general case, the inhomogeneous initial distribution of reactants, and avoid application of model concepts of a reaction system structure (i.e., of the structure of reactants and their molecular mobility). Slowing of multistage reaction kinetics as compared to the kinetics of elementary stages is established and physically interpreted. To test approximations (point approximation) used to develop a universal kinetic law, a widely employed specific model of spherical particles with isotropic reactivity diffusing in solution is applied. With this particular model as an example, ultimate kinetics of chemical conversion of reactants is investigated. The question concerning the depths of chemical transformation at which long-term asymptotes are reached is studied.

Numerical simulation of diffusion‐controlled nucleation and growth of porphyroblasts

AbstractThis article presents the conceptual basis for a new numerical model of diffusion‐controlled nucleation and growth of porphyroblasts, describes its implementation, and illustrates the range of crystallization behaviours encompassed by it. The model differs from previous efforts principally in its ability to track explicitly the evolution in time and space of the chemical affinity for reaction in the intergranular medium, which provides a more accurate characterization of nucleation suppression in the vicinity of pre‐existing crystals and of growth suppression due to competition for nutrients among neighbouring crystals. Critical new features of these numerical simulations include: maintenance of local equilibrium for fluid in contact with reactants or products; persistence of reactants until they are eliminated by dissolution due to reaction progress and local diffusive flux; nucleation rates that vary as the local reaction affinity evolves; complex initial distributions of reactants if desired; and the flexibility to encompass any porphyroblast‐forming reaction for which changes in free energy as a function of time and temperature are specified. Model results reveal that radial growth rates remain proportional to the square‐root of time in diffusion systems buffered by persistent reactants; they document the interchangeable effects of diffusivity, porosity, and solubility on material fluxes and thus growth rates; and they illustrate the offsetting textural effects of rates of diffusion, nucleation, and heating. The initial distribution of reactants is found to exert a first‐order effect on crystal size distributions, confirming their limited utility for diagnosing crystallization mechanisms. These numerical simulations yield novel and rigorous confirmation of the textural effects of nucleation‐site saturation and variation in interconnected porosity, and reproduce with high fidelity much of the textural and chemical information gathered from natural specimens.

Stochastic chemical reactions in microdomains

Traditional chemical kinetics may be inappropriate to describe chemical reactions in microdomains involving only a small number of substrate and reactant molecules. Starting with the stochastic dynamics of the molecules, we derive a master-diffusion equation for the joint probability density of a mobile reactant and the number of bound substrate in a confined domain. We use the equation to calculate the fluctuations in the number of bound substrate molecules as a function of initial reactant distribution. A second model is presented based on a Markov description of the binding and unbinding and on the mean first passage time of a molecule to a small portion of the boundary. These models can be used for the description of noise due to gating of ionic channels by random binding and unbinding of ligands in biological sensor cells, such as olfactory cilia, photoreceptors, hair cells in the cochlea.

Nonclassical kinetics of an elementary A+B-->C reaction-diffusion system showing effects of a speckled initial reactant distribution and eventual self-segregation: experiments.

We demonstrate here the implementation of an experimental system suitable for the study of the diffusion limited A+B-->0, nonclassical reaction behavior. Using a combination of a fluorescent calcium indicator and a calcium ion which is initially "caged," a pulse of near-UV light initiates the reaction which is followed as product formation vs time. Sensitive dependence on the initial reactant distribution is observed through patterns in the uncaging UV light profile. In one case, the reaction progress passes through two nonclassical time regimes, one due to roughness originating from laser speckles, followed by one consistent with the three-dimensional Zeldovich rate of (1/rho(A)-1/rho(A0)) approximately t(3/4), with features matching Monte Carlo simulations on this initial distribution. This behavior is contrasted with reactions initiated by a homogeneous source which induces random initial reactant distributions, though both systems seem to approach the asymptotic limit of self-segregation of reactants.

Observation of Laser Speckle Effects in an Elementary Chemical Reaction

AbstractAn experimental demonstration is shown for non-classical reaction kinetics in a homogeneous system with an elementary reaction, A+B→C. Sensitivity to the initial distribution of reactants is observed, along with a new reaction-kinetics regime which is a direct consequence of speckles in the laser beam. The long-time regime gives the first experimental demonstration of the asymptotic self-segregation (“Zeldovich”) effect, in spite of the non-random, speckled initial distribution of reactant B. Monte-Carlo simulation results are consistent with the experiments, and spatial analysis of these results correlates the excess of long-wavelength components in the initial reactant distribution with an anomalous slowing of the reaction progress.

Experimental Chemical Kinetics of an A+B→0 System Showing Non-Classical Effects of Initial Reactant Distribution

AbstractWe demonstrate here the implementation of an experimental system suitable for the study of the diffusion-limited A+B→0, non-classical reaction behavior. Using a combination of a fluorescent calcium indicator and a calcium ion which is initially “caged”, a pulse of near-UV light initiates the reaction which is followed as product formation vs. time. Sensitive dependence on the initial reactant distribution is observed through patterns in the uncaging UV light profile. The reaction progress passes through two non-classical time regimes, one due to roughness originating from laser speckles, followed by one exhibiting the three-dimensional Zeldovich rate of (1/ρA − 1/ρA0) ˜ t3/4, with features matching Monte Carlo simulations on this initial distribution.

Kinetics of nonstationary, single species, bimolecular, diffusion-influenced irreversible reactions

The extension to nonstationary situations of the statistical nonequilibrium thermodynamic theory of diffusion-influenced reactions is used to calculate the kinetics of the single species bimolecular chemical reactions. The method is based on the calculation of coupled dynamic equations for the average concentration and the radial distribution function. In particular, a detailed analysis is performed for the case of the reaction taking place in a one dimensional infinite medium. The single species bimolecular reactions (i.e., annihilation and coagulation) are found to have the same radial distribution function at all times in the low density limit, which implies that these reactions belong to the same spatial universality class under this criterion but not under the nearest-neighbor distance criterion. The rate of reaction depends on just one initial condition: the initial distribution of reactants, via g(r,0). For higher densities, the behavior is not universal and depends on the initial concentration of reactants, falling within the same universality class only if the initial concentration in the coagulation reaction is twice that of the annihilation reaction. This result agrees with what has previously been discussed in the literature using different approaches. The theory is checked against Monte Carlo simulations for the one dimensional case.

Diffusion-Limited Binary Reactions: The Hierarchy of Nonclassical Regimes for Correlated Initial Conditions

We predict hierarchies of kinetic regimes and crossover conditions for elementary A + A - 0 and A + B 0 batch reactions from a nonclassical reaction4iffusion formalism that includes spatial fluctuations. This paper addresses the case of a spatially correlated initial distribution of reactants; random initial conditions have been discussed in another paper. The hierarchies of kinetic regimes are even richer than those found for random initial conditions, with the particular hierarchy determined by the tightness of the initial correlations. In most cases there is one regime (which we call a burst) where the decay of the densities in the A + B reaction is even faster than in the A + A reaction. We confirm our predictions via detailed numerical simulations.

Nonclassical crossovers in binary reactions in one-dimensional systems

We examine the crossover behavior for the A + A → O and A + B → O reactions, which are highly nonclassical for a random initial distribution of reactants. We find that both reactions quickly depart from the classical behavior to a “depletion zone regime”, eventually reaching the segregated asymptotic behavior. We give the corresponding scaling laws for the crossover time and density. Finite size effects are also discussed. Computer simulations verify the scaling laws and provide the scaling coefficients.

Diffusion-limited binary reactions: the hierarchy of nonclassical regimes for random initial conditions

We derive a hierarchy of kinetic regimes and crossover conditions for elementary A + A → 0 and A + B → 0 batch reactions from a nonclassical reaction—diffusion formalism that includes spatial fluctuations. This paper addresses the case of a spatially random initial distribution of reactants; correlated initial conditions are discussed in another paper. For low dimensions ( d⩽2) we find that both A + A → 0 and A + B → 0 reactions depart (swiftly) from the classical behavior toward a “depletion zone regime” with non-Hertzian nearest neighbor distributions and nonclassical rate laws. Eventually the (A – B) density difference fluctuations take over in the A + B → 0 case and lead to the segregated Ovchinnikov—Zeldovich asymptotic behavior with its peculiar rate laws. We give scaling laws for the crossover time and crossover density, with explicit dependencies on the initial density and on the dimension. Similarly, in three dimensions the crossover from the classical to the segregated Zeldovich regime is derived for the A + B → 0 reaction. Finite size effects differ significantly for the segregated and nonsegregated regimes. In the former case we obtain a relation between aggregate sizes and lattice sizes. Monte Carlo simulations bear out the scaling laws and provide the scaling coefficients.

A stochastic approach to the theory of intramicellar kinetics. II. Master equation for reversible reactions

In this paper we focus on a particular class of intramicellar kinetic processes: reversible reactions of molecularity two taking place in the interior (or perhaps the immediate vicinity) of a micellar assembly. We formulate a stochastic master equation to describe an ensemble of compartmentalized, distributed systems wherein the (only) microscopic chemical event is a photoinduced, reversible, bimolecular reaction. Then, for an initial distribution of reactants assumed to be Poissonian, we calculate and characterize the overall (macroscopic) dynamics displayed by such a system. The resulting kinetic description is compared with the behavior found previously for the simpler case of irreversible, intramicellar kinetic processes. Finally, we introduce and then investigate the notion of an ’’apparent’’ equilibrium constant Q for a reversible reaction taking place in a compartmentalized, distributed system and compare this quantity with the ’’canonical’’ equilibrium constant K obtained for the (same) reversible reaction carried out in bulk, homogeneous solution. The full behavior of Q=Q(K) is explored numerically, and we prove analytically that Q∼K2 in the small K limit and that Q is independent of K in the limit of large K. The experimental consequences of the theory are dicussed in some detail.

Luminescence During Intermittent Electron Bombardment

The dynamic luminescence characteristics of five samples of commercial phosphors excited intermittently by electron bombardment have been studied. The decay of artificial willemite follows closely a simple exponential law, except for a rapid fall during the first three milliseconds after bombardment, the prominence of this component increasing with the current density. An attempt is made to correlate the observed dependence of build-up rate on current density with the phenomenon of current-saturation under continuous bombardment. A sample of calcium tungstate follows changes in the intensity of excitation within 100 microseconds, the resolving time of the apparatus. Two samples of ZnS·Ag and one sample of ZnS·CdS·Ag decay more rapidly the higher the intensity of excitation. The initial rate for ZnS·Ag is greater than would be predicted for a simple bimolecular reaction, and cannot plausibly be explained by non-uniformity in the initial distribution of reactants in a bimolecular reaction. The ZnS·CdS·Ag decays initially according to a simple bimolecular law.