Concentration dependence of diffusion-controlled processes among stationary reactive sinks

A multiple scattering theory of competition effects in diffusion-controlled reactions are presented. We consider a random array of stationary sinks which react with a density field of another reactant. Using the radiation boundary condition to describe the reaction at the surfaces of the sinks, we treat the modification of the density field due to reaction with sinks exactly. By keeping only the most divergent terms in a given order of scattering and summing them, we obtain the rate constant as a function of the sink concentration in the steady state. We also calculate the concentration-dependent diffusion constant of the density field. Both the rate and diffusion constants have nonanalytic behavior in the sink concentration.Alfred P. Sloan Foundation Fellow 1976–1980; John Simon Guggenheim Memorial Fellow 1979–1980.     

Diffusion-controlled reactions among stationary sinks

We summarize results of some of our calculations in diffusion-controlled reaction theory. We derive the transport equation describing a diffusing species which can react with a set of randomly distributed spherical sinks. Both the form of the transport equation and the dependence on sink volume fraction of the reaction rate and the effective diffusion coefficient are discussed.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.Supported in part by the National Science Foundation under Grant No. CHE-78-07849.     

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.     

Diffusion-controlled processes among partially absorbing stationary sinks

A general linear response theory is presented to calculate the zero-wavevector and zero-frequency reaction rate coefficient for particles diffusing into absorbing spheres. Allowance is made for possible incomplete particle absorption. A Faxén-like theorem for chemical reactions is derived. The problem is solved completely for a simple regular array of sinks. Exact analytic expressions for the rate coefficient as a function of sink volume fraction are obtained for the sc and fcc lattices. The case of a disordered array of sinks is also considered and the leading order nonanalytic density dependence of the rate coefficient is calculated. In both cases an increase in the rate coefficient with sink density in a local region of the system is found. The general formalism is extended to examine the modification to the particle diffusion coefficient due to the presence of the spheres. For regular arrays of spheres, the mean field result is reproduced.Research supported in part by a grant from the National Research Council of Canada.     

Rates of diffusion controlled reactions for one-dimensionally-moving species in 3D space

ABSTRACT

The asymmetry in diffusion dimensionality between self-interstitial atom (SIA) clusters and vacancies is a fundamental feature of irradiation damage in crystals, leading to a defect buildup imbalance that manifests itself as measurable dimensional and mechanical property changes. It is well known that, while vacancies and mobile vacancy clusters diffuse in a three-dimensional (3D) fashion, SIA clusters perform one-dimensional motion along mostly rectilinear trajectories. Despite this, a complete set of kinetic coefficients, including coagulation reaction rates and sink strengths, does not exist for 1D-moving objects. In this paper, we derive analytical expressions for these coefficients from continuum diffusion theory particularised to 1D motion. Moreover, we carry out kinetic Monte Carlo simulations of numerical replicas of the geometry of diffusing particles and sinks to validate the proposed solutions. Our simulations, which are conducted entirely independently from the analytical derivations, reveal excellent agreement with the proposed expressions, adding confidence to their validity. We compare the 1D and 3D cases and discuss their relevance for kinetic codes for damage accumulation calculations.     

Hybrid Isothermal Model for the Ferrohydrodynamic Chemically Reactive Species

A hybrid isothermal model for the homogeneous-heterogeneous reactions in ferrohydrodynamic boundary layer flow is established. The characteristics of Newtonian heating and magnetic dipole in a ferrofluid due to a stretchable surface is analyzed for three chemical species. It is presumed that the isothermal cubic autocatalator kinetic gives the homogeneous reaction and the first order kinetics gives the heterogeneous (surface) reaction. The analysis is carried out for equal diffusion coefficients of all autocatalyst and reactions. Heat flux is examined by incorporating Fourier's law of heat conduction. Characteristics of materialized parameters on the magneto-thermomechanical coupling in the flow of a chemically reactive species are investigated. Further, the heat transfer rate and friction drag are depicted for the ferrohydrodynamic chemically reactive species. It is evident that the Schmidt number has increasing behavior on the rate of heat transfer in the boundary layer. Comparison with available results for specific cases is found an excellent agreement.     

Population persistence in weakly-coupled sinks

We consider a single species population obeying a saturated growth model with spatial diffusion taken into account explicitly. Strong spatial heterogeneity is considered, represented by a position dependent reproduction rate. The geometry of the problem is that of two patches where the reproductive rate is positive, surrounded by unfavorable patches where it is negative. We focus on the particular case where the population would not persist in the single patches (sinks). We find by means of an analytical derivation, supplemented by a numerical calculation, the conditions for the persistence of the population in the compound system of weakly connected patches. We show that persistence is possible even if each individual patch is a sink where the population would go extinct. The results are of particular relevance for ecological management at the landscape level, showing that small patches may harbor populations as long as the connectivity with adjacent patches is maintained. Microcosmos experiences with bacteria could be performed for experimental verification of the predictions.     

Radiation enhanced diffusion in face centered cubic cu-zn alloys

The enhancement of diffusion by neutron irradiation has been investigated on a Cu-36 percent Zn alloy for various neutron fluxes and irradiation temperatures by means of in-pile measurements of electrical resistivity. For fresh samples the diffusion rate depends on temperature with an activation energy of 0.35 eV. During repeated irradiations the diffusion rate decreases and becomes nearly temperature independent. The variation of the concentration of interstitials and vacancies with irradiation time has been numerically calculated for various neutron fluxes, irradiation temperatures and sink concentrations. A comparison of the experimental and theoretical results shows that the point defects annihilate in fresh samples mainly by pair recombination and in samples which had been repeatedly cycled by pair recombination and at fixed sinks. Point defect clusters acting as sinks are created during the course of the irradiation as shown by electron microscope investigations. The radiation enhanced diffusion rate was found to depend on interstitials only, the activation energy of which was determined to 0.70 eV.     

On the theory of void formation during irradiation

A simple theory of the swelling of materials subjected to high energy particle irradiation is developed. Chemical reaction rate equations are used as a basis. Point defects, interstitials and vacancies, are assumed to be produced randomly throughout the solid. They move by random walk through the material until they cease to exist either by recombination with the opposite type of defect or by incorporation into the crystal at sinks such as dislocations, grain boundaries and voids. The rate equations for interstitials and for vacancies, which are coupled via the recombination term, are solved for steady state conditions under irradiation. Defect concentrations, supersaturations, recombination and total sink annihilation rates are obtained in terms of the production rate, sink annihilation probabilities, jump frequencies and thermal equilibrium concentrations of defects. The swelling rate is derived using sink annihilation probabilities at three principally different types of sinks, i.e. voids, sinks which have a bias with regard to the annihilation of interstitials and vacancies (such as dislocations), and sinks with no bias. The defect annihilation probabilities at void, precipitate, dislocation and grain boundary sinks are estimated by using a cellular model and solving the diffusion equation for geometries approximating that of the cells, e.g. a concentric sphere around a void. The relative effects of different types of sinks, i.e. the microstructure, on the swelling rate is discussed. The swelling rate is integrated to give swelling-time or swelling-dose relations, making some simplifying assumptions about the changes in the sink structure as the irradiation proceeds. It is shown that the relation obtained is rather sensitive to the type of assumptions made.     

Optimal reaction time for surface-mediated diffusion

We present an exact calculation of the mean first-passage time to a small target on the surface of a 2D or 3D spherical domain, for a molecule performing surface-mediated diffusion. This minimal model of interfacial reactions, which explicitly takes into account the combination of surface and bulk diffusion, shows the importance of correlations induced by the coupling of the switching dynamics to the geometry of the confinement, ignored so far. Our results show that, in the context of interfacial systems in confinement, the reaction time can be minimized as a function of the desorption rate from the surface, which puts forward a general mechanism of enhancement and regulation of chemical and biological reactivity.     

A Spectral Approach to Survival Probabilities in Porous Media

We consider a diffusive process in a bounded domain with heterogeneously distributed traps, reactive regions or relaxing sinks. This is a mathematical model for chemical reactors with heterogeneous spatial distributions of catalytic germs, for biological cells with specific arrangements of organelles, and for mineral porous media with relaxing agents in NMR experiments. We propose a spectral approach for computing survival probabilities which are represented in the form of a spectral decomposition over the Laplace operator eigenfunctions. We illustrate the performances of the approach by considering diffusion inside the unit disk filled with reactive regions of various shapes and reactivities. The role of the spatial arrangement of these regions and its influence on the overall reaction rate are investigated in the long-time regime. When the reactivity is finite, a uniform filling of the disk is shown to provide the highest reaction rate. Although the heterogeneity tends to reduce the reaction rate, reactive regions can still be heterogeneously arranged to get nearly optimal performances.     

Phase equilibria and critical phenomena in closed reactive systems

It is shown that the interplay between chemical reactions and criticality gives rise to some novel phenomena manifested both in a change of critical indices and in some pecularities in the course of chemical reactions. To cite a single example, one can mention the existence of a single point on the hypersurface of the diffusion instability where the slowing down of chemical reactions occurs. The requirements for phase separation in reactive systems are illustrated on simple models of a nonelectrolytic binary mixture and a ternary mixture including electrolytes. The general criterion for the existence of azeotropic points and the upper (lower) critical solution temperatures in reactive systems is formulated. The influence of a chemical reaction on the form of the solubility curve near the melting point in a binary and a dilute ternary mixture is analyzed in detail. A new general approach is formulated to the decay of metastable state in reactive systems. Finally, some possible experimental verifications are considered.This article is a tribute to my dear colleague C. Domb, who has done so much in the understanding of phase transitions and critical phenomena. It is a great pleasure to contribute to this volume honoring him on the occasion of his official retirement.     

Real time chemical dynamics at surfaces

It is a major goal in surface science to make movies of molecules on surfaces, in which the reaction of the molecules on the surface can be followed on a femtosecond time scale, with sub-nanometer resolution. By moving the actors (the molecules) to precisely determined positions on the stage (the surface) at some well-defined moment in time, and subsequently making a space- and time-resolved documentary of what happens next, we would be able to understand the reactive interactions between molecules on surfaces in the greatest possible detail. This would enable us to set the stage and bring together the actors in such a way as to produce the chemical outcomes our society needs, by improving existing catalysts and designing novel catalysts, and by engineering novel reactions on surfaces. Any future director of such movies needs to know which techniques (i.e., which theoretical and experimental methods) hold promise for movie making, what has been done with these techniques, and what can be done with appropriate extensions. The methods we discuss are: (i) the time-dependent wave packet method, which is a theoretical method for simulating molecule–surface reactions with sub-nanometer resolution on a femtosecond time scale, (ii) molecular beam experiments, which allow detailed investigation of the molecule–surface interaction at a molecular level, and (iii) time-resolved laser pump–probe experiments, which allow reactions to be studied with femtosecond resolution. In particular, we discuss (i) theoretical studies of the dissociation reaction of hydrogen on metal surfaces, the reactive system presently understood at the greatest level of detail, (ii) the reactive and non-reactive scattering of heavy diatomics (NO,CO) from metal surfaces, and (iii) the competition between reaction of coadsorbed CO with O and desorption of CO, again on a metal surface. We examine possibilities to extend these methods to make movies at the desired level of detail. We also discuss which reactions are likely to provide good material for plots of movies that will be exciting for future generations of surface scientists.     

Theoretical investigation of optical patterning of monolayers with subwavelength resolution

We formulate a model of monolayer patterning via optically-controlled chemical reactions, with the goal of beating the diffraction limit in photolithography. We consider the use of the proven technique of STimulated Emission Depletion (STED) to selectively place a handful of molecules in a reactive excited state. We show that repeated optical excitation has a greater effect on pattern formation than increasing the reaction rate, auguring well for experimental work. We also consider optically-controlled deposition of a soluble species via STED, and show that even for very large concentrations and excited state lifetimes the full width at half maximum of the features formed is robust against the effects of diffusion and saturation.     

Reversible A↔B reaction–diffusion process with initially mixed reactants: Boundary layer function approach

The reversible A?B reaction–diffusion process, when species A and B are initially mixed and diffuse with different diffusion coefficients, is usually considered as a diffusion-limited process. In this work the reaction rate in such a process is investigated using the boundary layer function method. It was shown that the reaction–diffusion process can be considered as a quasi-equilibrium process. Despite this fact the contribution of the changes in the species concentration of the reaction is comparable to that of the diffusion. Moreover, the ratios of the reaction and diffusion contributions are independent of time and coordinate. The dependence of the reaction rate on the initial species distribution is analyzed. It was demonstrated, for the first time, that the number of the reaction zones is determined by the initial conditions and changes with time. Also the asymptotic long-time behavior of the reaction rate depends on the initial species distribution.     

Interfacial reaction kinetics

We study irreversible A-B reaction kinetics at a fixed interface separating two immiscible bulk phases, A and B. Coupled equations are derived for the hierarchy of many-body correlation functions. Postulating physically motivated bounds, closed equations result without the need for ad hoc decoupling approximations. We consider general dynamical exponent z, where is the rms diffusion distance after time t. At short times the number of reactions per unit area, , is 2nd order in the far-field reactant densities . For spatial dimensions dabove a critical value , simple mean field (MF) kinetics pertain, where Q_b is the local reactivity. For low dimensions , this MF regime is followed by 2nd order diffusion controlled (DC) kinetics, , provided . Logarithmic corrections arise in marginal cases. At long times, a cross-over to 1st order DC kinetics occurs: . A density depletion hole grows on the more dilute A side. In the symmetric case (), when the long time decay of the interfacial reactant density, , is determined by fluctuations in the initial reactant distribution, giving . Correspondingly, A-rich and B-rich regions develop at the interface analogously to the segregation effects established by other authors for the bulk reaction . For fluctuations are unimportant: local mean field theory applies at the interface (joint density distribution approximating the product of A and B densities) and . We apply our results to simple molecules (Fickian diffusion, z=2) and to several models of short-time polymer diffusion (z>2). Received 8 June 1998 and Received in final form 10 September 1999     

Generation of thermal donors in silicon: Effect of self-interstitials

The generation of low-temperature thermal donors (TD) in silicon is sensitive to the sample cooling rate (from the anneal to room temperature) and the ambient (air or vacuum). This effect is most clearly pronounced in the case of annealing at 500°C, is noticeable at 480°C, and is practically undetectable at 450°C. The results are interpreted satisfactorily as being due to the TD generation becoming enhanced in the presence of silicon self-interstitial (Si_I) atoms. These atoms are emitted by thermal donors, to be subsequently absorbed by sinks, particularly the sample surface and grown-in microdefects (vacancy voids). When annealing in a vacuum, the surface acts as the main sink. If the anneal is done in air, this sink is passivated as a result of oxidation and/or contamination, with voids becoming the main sinks; as a result, the concentration of Si_I atoms increases substantially and the generation rate is enhanced. Rapid cooling brings about a partial passivation of the voids (as a result of their becoming decorated by rapidly diffusing impurities) and an additional enhancement of the generation rate. The calculated rate curves obtained within this model are well fitted to the experiment.     

Chemical reaction and diffusion: A comparison of molecular dynamics simulations with continuum solutions

Molecular dynamics simulations using the Lennard-Jones energy potential are compared with continuum solutions of reaction and diffusion in a dilute gas. The reaction model is a passive one in which high-energy bath atoms create a species, at dilute concentrations, which may have a very fast consumption reaction. This construction is designed based on typical fast reaction pathways involved in the fuel breakup in a hydrocarbon flame. Using reaction rates and diffusivities obtained from the molecular simulations allows the continuum solution to describe the reactive atom density spatial distribution with good accuracy. Based on this agreement, it is possible to estimate which reaction rates will produce negligible diffusive spreading, and hence, which species might be assumed to be in chemical equilibrium in continuum reacting flow calculations.     

非中性电解液的光子相关光谱

本文研究了非中性电解液光散射动态特性问题,导出了一元非中性电解液和二元非中性电解液的光子相关光谱。结果表明:它们的光子相关光谱是时间衰减谱,衰减系数同扩散系数只差因子q~2,微弱的非中性使利用光子相关光谱测得的扩散系数显著增加,且与散射角的关系更为密切。对于10~(-3)molar的稀薄电解液,1%的非中性就使扩散系数增大10~2～10~3倍以上。最后,本文对造成非中性效应的物理机制进行了分析。     

From atomistic lattice-gas models for surface reactions to hydrodynamic reaction-diffusion equations

Atomistic lattice-gas models for surface reactions can accurately describe spatial correlations and ordering in chemisorbed layers due to adspecies interactions or due to limited mobility of some adspecies. The primary challenge in such modeling is to describe spatiotemporal behavior in the physically relevant "hydrodynamic" regime of rapid diffusion of (at least some) reactant adspecies. For such models, we discuss the development of exact reaction-diffusion equations (RDEs) describing mesoscale spatial pattern formation in surface reactions. Formulation and implementation of these RDEs requires detailed analysis of chemical diffusion in mixed reactant adlayers, as well as development of novel hybrid and parallel simulation techniques. (c) 2002 American Institute of Physics.