Diffusion-limited Model Research Articles

NUMERICAL STUDY ON COMBUSTION OF A LIQUID FUEL DROPLET

Ignition and combustion of a liquid fuel droplet injected into a hot and highly pressurized combustor are investigated numerically, including the effect of internal circulation. Ignition delay times of droplets predicted with internal circulation are almost the same as those from the diffusion limit model insofar as ignition takes near the droplet. Combustion regime maps are drawn to classify the droplet combustion phenomena according to the configuration and location of the flame with respect to initial Reynolds numbers and the surrounding gas temperatures.

A crystal growth rate of equation for ancient dolomites; evidence for millimeter-scale flux-limited growth

ABSTRACT A linear correlation exists between the width of cathodoluminescent zones (CL) and the size of individual crystals in the Seroe Domi Fm (Miocene, Netherlands Antilles), Burlington-Keokuk Fm (Mississippian), Fort Payne Fm (Mississippian), and Saluda Fm (dovician). If individual zones, correlated by CL character and stratigraphy, are isochronous, then the width of a zone is directly proportional to the rate of crystal growth, Linear correlation coefficients for between-zone thickness and crystal size in individual thin sections range from 0.98 to 0.41, indicating a size dependence governing individual dolomite crystal growth rates. Mononuclear, polynuclear, spiral dislocation, and diffusion-limited crystal growth models operating under homogeneous or inhomogeneous kinetic conditions were considered as possible crystal growth mechanisms. Of the models considered, only mononuclear growth may result in a correlation between zone width and crystal radius in chemically homogeneous systems. However, this relationship is not linear. Any of the growth laws considered may produce a linear size-dependent growth rate when there are temporally constant differences between crystals in the kinetic factors governing growth. Differences in the kinetic factors governing growth could arise from stable variations in the chemistry of pore fluids related to permeability variations in the rock. The radius-rate data give a crystal growth rate equation of the form dr/dr = kr. This equation is consistent with a model wherein the rate of crystal growth is determined by stable variations in the chemistry of pore fluids. This implies that the kinetics of dolomite crystal growth is flux limited and therefore no slower than many diagenetic reactions. The data also support the hypothesis that thin-section-scale variations in dolomite crystal size can result from variations in crystal growth rate.

Nucleation model for diffusion-limited coalescence with finite reaction rates in one dimension

We study the diffusion-limited coalescence model $A+A\ensuremath{\rightarrow}A$ with nucleation and a finite reaction rate in one dimension for the particle density decay by means of a Monte Carlo simulation and analytic modeling. In our model, one or more particles in a lattice site act as a nucleus for the particles that diffuse into the site without reacting. The master equation governing the time evolution of local particle number and the rate equation for the particle density are derived. We present an analytic approach for the early time regime (reaction-controlled limit), which is strongly dependent on the initial particle density. In this regime, the particle density decays faster than the classical (or second-order reaction) limit and lower than the exponential decay (or first-order reaction) limit. For the long time regime the diffusion-controlled limit is recovered. We show that the intermediate regime can be obtained as an interpolation between the initial decay and the diffusion limit. The numerical integration results from the analytic approach are compared with computer simulations.

Diffusion-limited self-discharge reaction in the Hubble space telescope battery

The self-discharge rate of aerospace flight-quality nickel-hydrogen batteries is limited by hydrogen diffusion within the nickel electrode active material. A diffusion-limited reaction model is developed which accounts for the observed self-discharge behavior. Effective hydrogen diffusion coefficients were calculated from the open-circuit self-discharge data on Hubble space telescope flight-quantified nickel-hydrogen cells and ranged from 1.12×10 −10 cm 2/sec at 25°C to 7.45×10 −12 cm 2/sec at 0°C.

Blue, green and 0.8 μm Tm 3+, Ho 3+ doped upconversion laser glasses, sensitized by Yb 3+

To obtain efficient upconversion laser glass, the optical properties of Tm 3+ and Ho 3+ were investigated in various glasses. Fluoride glass was selected as base glass for upconversion. The efficient upconversion fluorescences corresponding to the 1G 4→ 3H 6 and 3H 4→ 3H 6 transitions of Tm 3+ and the 5S 2→ 5I 8 transition of Ho 3+ were observed in Yb 3+-Tm 3+ and Yb 3+-Ho 3+ doped aluminozircofluoride glasses excited at 980 nm. The very stronge blue and green emission light can be observed visually. The upconversion processes observed were two-photon processes for 3 F 4→ 3 H 6, 5 S 2→ 5 I 8 transitions and three-photon processes for the 1G 4→ 3H 6 transition and can be described by a rate equation model. The energy transfer and energy back-transfer were analyzed in Yb 3+-Tm 3+ and Yb 3+-Ho 3+ systems. The relationship between emission intensity, pumping intensity and dopant concentrations is described using a rate equation model and shows good agreement with experiments. The dynamics of excited state ( Yb 3+2 F 5 2 ) is also analyzed with the diffusion-limited model based on Yokota-Tanimoto theory.

Quantum diffusion and localization of positive muons in superconducting aluminum.

The diffusion of positive muons was studied in the normal and superconducting states of aluminum. Large differences were observed, indicating that the diffusion mechanism depends sensitively on the interaction between the muon and the conduction electrons. The samples were high-purity aluminum doped with controlled amounts of lithium. The lithium impurities act as traps for the muons, leading to a loss of muon polarization. However, the time and temperature dependencies of the muon depolarization function G(t) could not be satisfactorily explained using a simple diffusion limited trapping model. In particular, in the superconducting state the measured G(t) requires a more complex model for a reasonable fit. These data can be described, in a first approximation, with two fractions, one corresponding to muons stopped in the defect potential created by the doping element and the other representing muons which diffuse freely without being trapped. In this model, which supports a microscopic theory by Kagan and Prokof'ev taking into account static as well as dynamic effects on the tunneling of positive particles in a conducting medium, a large fraction of the muons is practically immobile below 0.3 K in doped samples. The data can also be fitted by assuming that the spatial distribution of doping elements leads to a distribution in trapping times and that the fraction of diffusing muons is described by a so-called stretched exponential time dependence. The relative merits of these two types of interpretation are discussed.

A modified computer model for the formation of porous silicon

A modified diffusion-limited model for the formation of porous silicon is proposed. The modified model retains the simplicity and effectiveness of the original model by R.L. Smith et al., yet provides a more accurate description of the porous Si formation. In particular, the modified model considers the probability of a reaction between the F − ions and the holes and is linked with the actual anodization conditions. The model explains why the anodic current density and HF concentration affect the formation of porous Si. The computer simulation results show a variation of porosity as functions of HF concentration and anodic current density, which has a trend closely matching that from the experimental results.

Fraction of hydrocarbon emission in thermal re-emission of hydrogen implanted into graphite

Isothermal re-emission of hydrogen, implanted into graphite with 3 keV H 2 + ions up to different fluences at room temperature has been measured at 450, 500 and 600°C by means of the elastic recoil detection (ERD) technique. It is found that as the time of the post-isothermal annealing increases the concentration of hydrogen retained at each implantation fluence decreases rapidly in the beginning and hereafter very gradually. The decay curves due to hydrocarbon molecule emission are obtained as the differences between the experimental decay curves and the decay curves due to hydrogen molecule emission, calculated by fitting the solution of the mass balance equations to the experimental ones at large annealing times, where thermal detrapping, retrapping and local molecular recombination are taken into account without hydrocarbon molecule formation. The ratios of the rate constants of hydrocarbon molecule formation to trapping are determined by best fitting of the calculated decay curves due to hydrocarbon molecule emission to the experimental ones obtained in the above procedure. The ratios determined and the process of hydrocarbon molecule formation are discussed on the basis of the diffusion-limited reaction model.

Middle ear gas exchange in rhesus monkeys.

The mechanism of gas exchange between the middle ear and tissue is fundamental to understanding middle ear physiology and pathophysiology. In this study, the middle ears of six rhesus monkeys were inflated on separate occasions with nitrogen (N2), carbon dioxide (CO2), and oxygen (O2), and middle ear pressures were recorded at defined times postinflation for up to 4 hours. From these data, rate constants governing the exchange of these gases were estimated and compared to those predicted under both diffusion and perfusion limitations. The results show that the rate constants for middle ear to tissue exchange of O2 and CO2 are consistent with a diffusion-limited process. In contrast, middle ear pressure did not decrease over the study period following introduction of N2 into the middle ear. This is interpretable as a much slower rate of N2 exchange than that predicted by either perfusion or diffusion-limited models calibrated to the O2 and CO2 rate constants. These results have significant implications for middle ear gas exchange and suggest that for relatively short observation periods, the behavior of middle ear pressure is controlled by experimentally established O2 and CO2 gradients.

Preparation of Ge quantum crystallites embedded in a-SiN x matrix by the PECVD method

Preparation of Ge quantum crystallites embedded in a-SiN x matrix was successfully achieved by the PECVD technique and followed thermal annealing treatment at 800°C. The microscopic heterogeneity of the as-deposited and thermal-annealed films were analyzed by the TEM and X-ray diffraction. We have found that substrate temperature is a critical parameter for the formation of Ge clusters. The temperature and time duration of annealing determine the size of Ge quantum crystallites. We are temporarily using the diffusion-limited growth model to explain the crystallization mechanism of this quantum material.

A model of diffusion-limited ice growth inside biological cells during freezing

A theoretical model for predicting the kinetics of ice crystallization inside cells during cryopreservation was developed, and applied to mouse oocytes, by coupling separate models of (1) water transport across the cell membrane, (2) ice nucleation, and (3) crystal growth. The instantaneous cell volume and cytosol composition during continuous cooling in the presence of glycerol were predicted using the water transport model. Classical nucleation theory was used to predict ice nucleation rates, and a nonisothermal diffusion-limited crystal-growth model was used to compute the resulting crystallization kinetics. The model requires knowledge of the nucleation rate parameters Ω and κ, as well as the viscosity η of a water-NaCl-glycerol solution as a function of both the composition and temperature of the solution. These dependences were estimated from data available in the literature. Cell-specific biophysical parameters were obtained from previous studies on mouse oocytes. A sensitivity analysis showed that the model was most sensitive to the values of κ and η. The coupled model was used to study the effect of cooling rate and initial glycerol concentration on intracellular crystal growth. The extent of crystallization, as well as the crystal size distribution, were predicted as functions of time. For rapid cooling at low to intermediate glycerol concentrations, the cells crystallized completely, while at high concentrations of glycerol, partial or total vitrification was observed. As expected, the cooling rate necessary for vitrification dropped with increasing glycerol concentration; when cells initially contained ∼7.5 M glycerol, vitrification was achieved independent of cooling rate. For slow cooling protocols, water transport significantly affected the results. At glycerol concentrations greater than ∼3 M, the final intracellular ice content decreased with increasing glycerol concentration at a fixed cooling rate. In this regime, the cooling rate at which a critical amount of ice was formed increased as more glycerol was used. When less than ∼3 M glycerol was initially present in the cell, an increase in glycerol concentration was predicted to cause an increase in the final intracellular ice content at a given cooling rate. In this regime, the critical cooling rate decreased with increasing glycerol concentration. These predictions clarify previous empirical observations of slow freezing phenomena.

On the universality of two-dimensional diffusion-limited deposition

A history dependent stick probability is introduced to the diffusion-limited deposition model. The exponents in the scaling laws are calculated. The universality class is also discussed.

Reduction-of-dimensionality kinetics at reaction-limited cell surface receptors

It has been suggested for several years that reactions between ligands and cell surface receptors can be speeded up by nonspecific adsorption of the ligand to the cell surface followed by two-dimensional surface diffusion to the receptor, a mechanism referred to as "reduction-of-dimensionality" (RD) rate enhancement. Most of the theoretical treatments of this and related problems have assumed that the receptor is an irreversibly absorbing perfect sink. Such receptors induce a depletion zone of ligand probability density around themselves. The reaction rate in this case (called "diffusion-limited") is limited only by the time required for ligands to diffuse through this depletion zone. In some cases, however, the receptor may be far from "perfect" such that a collision with a ligand only rarely leads to binding. Receptors then do not create significant local depletion zones of ligand probability density, and the reaction rate becomes strongly affected by the (small) probability of reaction success per diffusive encounter (the "reaction-limited" case). This article presents a simple theory of RD rate enhancement for reaction-limited receptors that are either reversible or irreversible binders. In contrast to the diffusion-limited theories, the reaction-limited theory presented here: (a) differs quantitatively from diffusion-limited models; (b) is simple and algebraic in closed form; (c) exhibits significant rate enhancement in some realistic cases; (d) depends strongly on the actual Brownian rather than pure diffusive nature of the ligand's motion; (e) depends (for irreversibly binding receptors only) on the kinetic rates (not just equilibria) of reversible adsorption to nontarget regions, in contrast to some previous approximate theories of reduction of dimensionality; and (f) is applicable to actual ligand/receptor systems with binding success probabilities at the opposite extreme from the perfect sink/diffusion-limited models.

Kinetics of DNA binding to electrically conducting polypyrrole films

The kinetics of 32 P end-labeled DNA binding from solution to electrochemically prepared polypyrrole (PPy) thick films are investigated. These data were found to fit a simple diffusion-limited binding model with low phenomenological activation energy (5 kcal/mol), consistent with diffusion-limited binding kinetics. These data suggest that the DNA binds to PPy without significant alteration of its structure. The binding of double-helical DNA to PPy was found to be dependent upon the presence of the electrooxidized positively charged surface functional groups on PPy. DNA adsorption kinetics are only moderately affected by changes in ionic strength and pH in the 5-9 range. A twofold increase in binding rate is observed for single-stranded DNA compared to the double-stranded DNA binding level

Facilitated transport of ethene through Nafion membranes. Part I. Water swollen membranes

Poly(perfluorosulfonic)ionomer membranes (Nafion N-117) were studied for the separation of ethene from ethane. Membranes in the sodium from were ion-exchanged in an aqueous 2 M solution of Ag +, and the flux of ethene was measured as a function of the partial pressure of ethene in the humidified feed gas. The experimental data were consistent with a carrier transport mechanism for a diffusion limited analytical model. The plot of the inverse facilitated ethene flux versus the inverse partial feed pressure was used to calculate the equilibrium constant for complexation of silver and ethene within the membrane. Effective diffusion coefficients for ethene were also calculated from the flux data. The facilitation factor for ethene with a humidified feed mixture of 50 mole% ethane and 50 mole% ethene at total atmospheric pressure was 170. The separation factor was 470. The permeability coefficient for ethene corresponded to 1000 barrer.

Averaged interdiffusion coefficient during solid-state amorphization reactions between metal layers.

We have found that diffusion-limited growth models with constant, averaged interdiffusion coefficients are inadequate as a description of the kinetics of the solid-state amorphization reactions in Ni/Zr multilayered composites. Although solid-state amorphization reactions appear to be diffusion controlled, the averaged interdiffusion coefficient appears to decrease as the reaction progresses. This observation provides some explanation for discrepancies in a number of measurements associated with these phenomena.

Effects of Variable Liquid Properties on Multicomponent Droplet Vaporization

A multicomponent droplet vaporization model, the Diffusion-Limit Model, is modified to account for the variation of liquid properties due to large temperature gradients as well as considerable concentration gradients within the droplet. The effects on the vaporization behavior are analyzed for an isolated biocomponent droplet consisting of heptane and dodecane. The results are presented for both moderate and high gas temperatures excluding combustion. During the vaporization process the liquid phase properties vary considerably. For example, the Lewis number changes by approximately one order of magnitude. The mass ratio of the liquid components seems to be rather sensitive to the variation of thermophysical property values, especially during the second half of the droplet lifetime, where about 50 percent of the droplet mass will still evaporate. The gas phase behavior is less affected by the use of constant liquid properties. For both gas temperature levels tested it was found that single component models cannot describe satisfactorily the whole vaporization process of multicomponent droplets. With regard to ignition the sharp rise of the vapor concentration in the beginning of the droplet vaporization is important. This behavior is caused by the more volatile component and cannot be achieved by the single component substitute.

Rate and extent of supercritical fluid extraction of additives from polypropylene: Diffusion, solubility, and matrix effects

AbstractAnalytical extraction of additives from polymers using a supercritical fluid (SFE) is a promising alternative to liquid extraction. Factors affecting SFE with carbon dioxide of Irgafos 168 and Irganox 1010 from commercial polypropylene have been studied, with analysis of extracts by capillary supercritical fluid chromatography (SFC). A diffusion limited extraction model was investigated by measuring the rate of SFE as a function of pressure, particle size, flow rate, and temperature. The rate of extraction was found to fit the sum of exponential decays; results were also consistent with an extrapolation procedure to obtain the total mass of additive without exhaustive extraction. Derived diffusion coefficients for the additive in polypropylene differed from literature values by approximately two orders of magnitude apparently because of swelling of the polymer by sorbed carbon dioxide. The variation of extraction rate with pressure and flow rate confirmed the solubility limitation of proposed model. The rate of extraction increased between 20 and 140°C, then fell away at higher temperatures with the onset of melting. © 1993 John Wiley & Sons, Inc.

Comparison of results and models of solid-phase epitaxial growth of implanted Si layers induced by electron- and ion-beam irradiation.

An outline of new and previously published results of solid-phase epitaxy of implanted Si layers induced below room temperature by electron irradiation in the transmission electron microscope is given. The basic features of the process (i.e., the dependence of epitaxial growth rate on temperature of the target, electron energy, dose rate of irradiation) are discussed and compared with the trends reported in the literature for ion-beam-induced epitaxy obtained in a higher-temperature range (typically 200\ensuremath{\le}T=400 \ifmmode^\circ\else\textdegree\fi{}C). The aim is to demonstrate that the fundamental characteristics of electron-beam-induced epitaxy are consistent with the extrapolation to low temperature and low mass of irradiating particles of the features of ion-beam-induced epitaxy, if appropriate assumptions on the kinetics of the defects responsible for the transformation are made. To better clarify this point it is shown that both a purely interface- and a purely diffusion-limited model of particle-induced epitaxy can in principle reproduce the very essential features of the process: the key point being the assumptions made on the dominant mechanism of defect reaction. In fact, while a linear-recombination scheme seems appropriate to explain the results of electron-induced epitaxy below room temperature, a dominant bimolecular-reaction mechanism seems necessary to reproduce the essential features of ion-induced epitaxy. Possible influences on electron-beam-induced epitaxy of electronic excitation or elastic collisions under the threshold for production of Frenkel pairs are also qualitatively discussed.

Diffusion-limited exciton fusion reaction in one-dimensional tetramethylammonium manganese trichloride (TMMC).

We present results of an experimental study on the reaction kinetics of a one-dimensional diffusion-reaction system, on a picosecond to millisecond time scale. Tetramethylammonium manganese trichloride (TMMC) is a perfect model system to study this problem. Time-resolved luminescence of TMMC has been measured over nine decades in time. The nonexponential shape of the luminescence decay curves depends strongly on exciting laser power. This is shown to result from a fusion reaction (A+A\ensuremath{\rightarrow}A) between photogenerated excitons, which is very well described by the diffusion-limited single-species fusion model for initial exciton densities 2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$ (fraction of the number of sites). At higher initial exciton densities the diffusion process, and thus the reaction rate, is significantly influenced by the heat produced in the fusion reaction. This is supported by Monte Carlo simulations.