Reaction-limited Regime Research Articles

A Scaling Theory for Number-Flux Distributions Generated during Steady-State Coagulation and Settling and Application to Particles in Lake Zurich, Switzerland

In this study, we extend the established scaling theory for cluster size distributions generated during unsteady coagulation to number-flux distributions that arise during steady-state coagulation and settling in an unmixed water mass. The scaling theory predicts self-similar number-flux distributions and power-law decay of total number flux with depth. The shape of the number-flux distributions and the power-law exponent describing the decay of the total number flux are shown to depend on the homogeneity and small i/ j limit of the coagulation kernel and the exponent κ, which describes the variation in settling velocity with cluster volume. Particle field measurements from Lake Zurich, collected by U. Weilenmann and co-workers ( Limnol. Oceanogr. 34, 1 (1989)), are used to illustrate how the scaling predictions can be applied to a natural system. This effort indicates that within the mid-depth region of Lake Zurich, clusters of the same size preferentially interact and large clusters react with one another more quickly than small ones, indicative of clusters coagulating in a reaction-limited regime.

In situ kinetic analysis of SiC filaments CVD

An in situ kinetic study of the SiC-based filament CVD from DCMS/H2 and DCMS/C3H6/H2 mixtures has been carried out. The results evidenced a low temperature regime (T<1150 °C) controlled by surface chemical reactions and a high temperature domain (T>1150 °C) governed by the diffusion of reactive species to the substrate (mass transfer regime). The addition of propene to the DCMS/H2 mixture leads to the decrease of the deposition rate, the increase of the activation energy (within the surface reaction limited regime) and the increase of the C/Si (at.) ratio. The dilution of DCMS in H2 promotes the co-deposition of free silicon. Conversely, the addition of propene results in lower amounts of free silicon or even traces of free carbon for high deposition temperatures. The increase of the C/Si (at.) ratio observed in presence of propene might be related to the inhibition of the deposition of Si-rich species derived from DCMS, through reactions with propene or related unsaturated hydrocarbons in the gas phase.

Phosphorus Doping in Si1-x-yGexCy Epitaxial Growth by Low-Pressure Chemical Vapor Deposition Using a SiH4–GeH4–CH3SiH3–PH3–H2 Gas System

Phosphorus doping in Si1-x-yGexCy (0≦x≦0.78, 0≦y≦0.016) epitaxial growth on Si(100) at 550°C by ultraclean hot-wall low-pressure chemical vapor deposition using a SiH4–GeH4–CH3SiH3–PH3–H2 gas system is investigated. The relationship among the Ge, the C fraction, the P concentration (CP), the deposition rate, and the deposition conditions in the P-doped Si1-x-yGexCy epitaxial growth under the surface reaction-limited regime is experimentally obtained, and is explained by the modified Langmuir-type adsorption and reaction scheme. The relationships among the carrier concentration, the CP, and the resistivity in the P-doped Si1-x-yGexCy for various Ge and C fractions are also presented. The carrier concentration of the P-doped Si1-x-yGexCy with low Ge and C fractions (x≦0.48 and y≦0.0046) is nearly equal to CP below approximately 2×1020 cm-3. With increasing Ge and C fractions, the film has electrically inactive P atoms independent of CP. The existence of C (y≧0.0048) in the film reduces the Hall mobility.

Computer simulations of diffusion-limited reactions

In many situations, it is possible to model a system by entities that diffuse and occasionally react upon encounter. Examples include chemical reactions, electron-hole recombination, annealing of defects in crystals, interacting excitons (solitons, photoexcited states, and so on), predator-prey models, and the spreading of ideas through social intercourse. For concreteness and without loss of generality, we refer to the interacting entities as particles. We discuss various ways to simulate diffusion-reaction systems. For simplicity, we restrict our discussion to simple reaction schemes, such as coalescence and annihilation, and to the case where the particles occupy a 1D lattice. It turns out that diffusion-limited kinetics deviates the most from the classical reaction-limited regime in d=1, so the focus on one dimension is reasonable. The generalization of the techniques discussed to more realistic models (off-lattice, higher dimensions, and so on), is straightforward.

Electrical characterization of ultrathin oxides of silicon grown by N2O plasma assisted oxidation

The ultrathin (2.0–3.5 nm) oxides of silicon have gained renewed importance in view of ultra large scale integration (ULSI) of the silicon devices. In the present investigation, the ultrathin oxides are grown on (100) oriented p-type single side polished silicon using N20 plasma assisted oxidation in a PECVD reactor at 200°C. The oxide growth as a function of oxidation time is studied. The oxidation growth conforms to the reaction limited regime. In order to understand the electrical quality of Si/ultrathin SiO2 interface, Al-thin SiO2-Si tunnel capacitors are fabricated and their capacitance-voltage (C-V) and current-voltage (I–V) characteristics are studied. The effect of annealing on these oxides (termed as “post oxidation annealing”) has also been studied. The C-V characteristics of tunnel capacitors with “as grown” oxide showed a frequency dependence, possibly due to the presence of large fast interface state density. These fast interface states are observed to decrease with increasing oxidation time. The tunnel capacitors that the oxides undergone “post oxidation annealing” (POA) at 350°C in N2 ambient for 20 minutes have shown practically no frequency dependence of the C-V characteristics; this observation along with the data on I-V characteristics confirms that POA reduces the interface state density considerably. The forward and reverse currents of POA capacitors are observed to decrease considerably indicating the reduction in the trap assisted tunneling transport process across the tunnel insulator.

Wet chemical etching of NiFe, NiFeCo AND NiMnSb for magnetic device fabrication

Thin film NiMnSb is found to be wet chemically etched in a range of solutions, including HNO 3, H 2SO 4:H 2O 2, FeCl 3 and HF, with activation energies in the range 10–23 kCal mol −1, i.e. a reaction-limited regime. The degree of sidewall undercut is a function of solution type. Both NiFe and NiFeCo can be etched in H 2SO 4 and HNO 3, with reaction-limited characteristics, while HCl at 25°C is completely selective for NiFe over NiFeCo, even for 7% Co alloys. The degree of sidewall undercut on NiFe is also a strong function of solution type.

Dynamic scaling of a reaction-limited decay process

A model of surface deconstruction in 1+1 dimensions is presented in this paper. The process of decay goes on only at surface sites in a reaction-limited regime and the statistical properties of surface sites are studied in comparison with the well-known results of the Eden growth. In particular, the dynamic behaviour of the short-time exponent β is investigated for large strips and differences between Eden growth and the present decay model are pointed out. It is shown how a simple scheme of site extraction can produce a multiple cluster decay whose secondary mechanical effects often arise from and overlap to a generic process of chemical erosion. The analysis of the particles released during deconstruction is related to experimental results on highly ordered pyrolythic graphite. Such an analysis can explain some discrepancies between continuum theory and a specific chemical disaggregation.

Derivation of an analytical model to calculate junction depth in HgCdTe photodiodes

An enhanced analytical model is derived to calculate the junction depth and Hg interstitial profile during n-on-p junction formation in vacancy-doped HgCdTe. The enhanced model expands on a simpler model by accounting for the Hg interstitials in the p-type, vacancy-rich region. The model calculates junction depth during both the initial, reaction-limited regime of junction formation and the diffusion-limited regime. It also calculates junction depth under conditions when the abrupt junction approximation of the simpler model fails. The enhanced model can be used to determine the limits of the annealing conditions and times for which the junction depth calculated analytically is valid. The decay length of interstitials into the p-type region estimated analytically places an upper bound on the grid spacing needed to accurately resolve the junction in a numerical simulation.

Evidence for a crossover region in the aggregation of PEO-coated polystyrene particle suspensions

The crossover region in the salt-induced aggregation processes occurring in polystyrene latex suspensions has been investigated by means of dynamic light scattering measurements. The possible existence of a region between reaction-limited and diffusion-limited regimes has been suggested in the last few years on the basis of various experiments and computer simulations, but no definitive agreement has been reached, since reaction-limited aggregation can be viewed as the initial regime crossing over towards diffusion-limited aggregation after a certain time and, in any case, in the long time limit. In the present work we have varied the overall interaction potential between particle monomers and thus the corresponding aggregation regime by varying the amount of adsorbed polymer [poly(ethylene oxide) (PEO), in this case] on the particle surface. The aggregation is induced by adding an appropriate simple salt (NaCl, 0.5 M) to the PEO-coated latex particles, at different polymer surface coverages, giving rise to different aggregation regimes, from fast aggregation in the absence of PEO, to slow aggregation at moderate PEO concentration, up to complete stabilization of the colloidal suspension at higher polymer concentration. In the present case, deviation from a reaction-limited regime extends for a period of time long enough to allow accurate analysis of the data. Comparison of the experimental data with the prediction of the Smoluchowski equation, modified by Tartaglia etal. (Phys. Rev. E, 1994, 50, 1649) to take into account the intermediate regime, is quite satisfactory and values of the parameters of the interaction potential between monomers have been evaluated. The present analysis gives support to the suggestion that slow, reaction-limited cluster aggregation is only a transient regime of the early stage of aggregation.

Reactive polymer membranes for ethylene/ethane separation

Olefin/paraffin separations by distillation are highly energy intensive. Facilitated transport, or reactive membranes have long been investigated as an alternative and/or complementary separation technology to conventional distillation. However, stability problems associated with facilitated transport membranes have been the primary obstacle in the development of commercial FT processes. In this paper, we report the development of a polymer membrane containing silver(I) ion that facilitates the transport of ethylene in the absence of solvent. Blends of ionically conductive and electrically conductive polymers were found to have the appropriate electronic environment to allow reaction of silver(I) ion and ethylene. The results for the ethylene/ethane separation were obtained with composite membranes of silver(I)-form Nafion ® and 2 wt% poly(pyrrole). Permeation measurements were performed with ethylene/ethane mixtures at total feed pressures ranging from 760 to 1900 mmHg and at temperatures of 40°C to 70°C. Pure gas permeability measurements were obtained at a total feed pressure of 1900 mmHg and temperatures of 30°C and 40°C. Ethylene/ethane separation factors with the silver(I)-form Nafion-poly(pyrrole) composite membranes increased from 8 to 15 as temperature decreased. Ethylene permeabilities increased from 0.2 to 1 Barrer over the temperature range of 30°C to 70°C. An ethylene/ethane mixed gas permeability ratio of about 2 was observed with non-reactive proton-form Nafion-poly(pyrrole) composite membranes. Ethylene permeation measurements as a function of membrane thickness suggested that the facilitated transport of ethylene approached the reaction-limited regime at membrane thickness of 5 μm. The complexation between ethylene molecules and silver(I) ions in Nafion-poly(pyrrole) composite membrane was observed with FTIR spectroscopy.

Modulated-beam studies of the layer-by-layer etching of GaAs(0 0 1) using AsBr3: identification of the reaction mechanism

Modulated-beam mass spectroscopy (MBMS) has been used to study the reaction mechanism of the layer-by-layer etching of GaAs(0 0 1) using AsBr3 under molecular beam epitaxy conditions. It is shown that GaBr is the main etching product and its “delay” time with respect to the incident AsBr3 flux exhibits a strong dependence on substrate temperature, changing from 12 ms in the reaction limited regime at 387°C to less than 0.5 ms at 560°C. Desorbing As-containing species appear to have very short surface lifetimes throughout the temperature range investigated and the additional As2 flux supplied from a solid source has no effect on the etching rate. The results suggest a reaction pathway where the rate limiting step is the formation or desorption of GaBr and not the decomposition of AsBr3.

Evolution of nanocluster ensembles: Computer simulation of diffusion and reaction controlled Ostwald ripening

The Ion Beam Synthesis (IBS) of nanoclusters can be controlled by the flux and the fluence of ions as well as by the implantation and/or annealing temperature. The evolution of the precipitates, mainly described by their particle radius distribution (PRD), the number of remaining particles and the critical radius, depends on these parameters. However, the evolution is expected to behave different for systems like SiO 2 and CoSi 2 clusters in Si, because they are characterized by quite different ratios of diffusion to reaction constants. Here we present a unified model of Ostwald ripening of nanoclusters, which describes the pure diffusion and reaction controlled limits as well as all intermediate, mixed processes. Expressing the exact solution of the adiabatic diffusion equation by multipole moments we derive the governing equation for the evolution of an ensemble of nanoclusters in the leading monopole approximation. Whereas in the diffusion controlled regime we have the well known infinite range 1 r interaction among the nanoclusters, 1 r 2 interactions come into play in the reaction limited regime. Computer simulations for ensembles of several thousands of nanoclusters demonstrate the evolution for typical cases.

Kinetics of fragmentation-annihilation processes

We investigate the kinetics of systems in which particles of one species undergo binary fragmentation and pair annihilation. In the latter, nonlinear process, fragments react at collision to produce an inert species, causing loss of mass. We analyze these systems in the reaction-limited regime by solving a continuous model within the mean-field approximation. The rate of fragmentation for a particle of mass x to break into fragments of masses y and x-y has the form ${\mathit{x}}^{\ensuremath{\lambda}\mathrm{\ensuremath{-}}1}$ (\ensuremath{\lambda}&gt;0), and the annihilation rate is constant and independent of the masses of the reactants. We find that the asymptotic regime is characterized by the annihilation of small-mass clusters, with the cluster density decaying as in pure annihilation and the average cluster mass as in pure fragmentation. The results are compared with those for a model with linear mass loss (i.e., with a sink rather than a reaction). We also study more complex models, in which the processes of fragmentation and annihilation are controlled by mutually reacting catalysts. Both pair and linear annihilation are considered. Depending on the specific model and initial densities of the catalysts, the time decay of the cluster density can now be very unconventional and even nonuniversal. The interplay between the fragmentation and annihilation processes and the existence of a scaling regime are determined by the asymptotic behavior of the average mass and of the mass density, which may either decay indefinitely or tend to a constant value. We discuss further developments of this class of models and their potential applications. \textcopyright{} 1996 The American Physical Society.

Multi-method determination of the fractal dimension of hematite aggregates

The utilisation of the fractal concept has greatly improved our ability to describe the structure of colloidal aggregates and our understanding of their formation mechanisms. We have determined the fractal dimension of KCl-induced hematite aggregates using three different approaches: (i) dynamic scaling, (ii) TEM image analysis and (iii) static light scattering (SLS). In the regime of diffusion limited aggregation ([KCl] > 100 mM), analysis of TEM images of 70 aggregates, whose longest dimension varies more than 40 fold, yields the lower limit of the fractal dimension as 1.68 ± 0.04. Direct scaling of the mean hydrodynamic size of the aggregates measured by photon correlation spectroscopy gives the upper limit of the fractal dimension as 1.87 ± 0.07. The value obtained by static light scattering as 1.83 ± 0.07. In the reaction limited regime, SLS measurements yield a fractal dimension of 2.2 ± 0.1. These results suggest the aggregates grow mainly through collision with other aggregates, rather than through the addition of primary particles.

Steady state of imperfect annihilation and coagulation reactions

We study the coagulation (A+A -, A) and annihilation (A+A -+ 0) reactions with input probability B and reaction probability p in a omdimensional latlice. In the steady state we find two different behaviours for the density of nearest-neighbour occupied sites r against the density of particles p. These behaviours correspond to the diffusion-limited regime (p + 0) and to the reaction-limited regime (p - 1). Using a scaling ansak for r against p we derive an approximation for p as a function of 6 and p that agrees well with Monte Carlo numerical results.

Growth Kinetics, Silicon Nucleation on Silicon Dioxide, and Selective Epitaxy Using Disilane and Hydrogen in an Ultrahigh Vacuum Rapid Thermal Chemical Vapor Deposition Reactor

Silicon nucleation on silicon dioxide and selective silicon epitaxial growth (SEG) were studied in an ultrahigh vacuum rapid thermal chemical vapor deposition (UHV‐RTCVD) reactor using 10% diluted in . Silicon was deposited on patterned Si (100) substrates over a pressure range of 10–100 mTorr and a temperature range of 650 and 850°C. Under these conditions, the growth rate ranged from 50 to 330 nm/minute, demonstrating compatibility with single wafer manufacturing throughput requirements. A pressure dependence in the activation energy in the surface reaction limited regime was observed and attributed to a variation in the steady‐state hydrogen coverage on the growing surface. The incubation time for loss of selectivity via Si nucleation on was found to increase at lower pressure and remained constant over the experimental temperature range. However, the incubation thickness defined as the film thickness that can be deposited before loss of selectivity occurs was found to increase both at low pressures and high temperatures. We show that a 100 nm thick epitaxial film can be grown selectively with no Cl addition at 750°C/10 mTorr.

Laser-induced chemical vapor deposition of titanium diboride

The CO2 laser-induced chemical vapor deposition process of titanium diboride (TiB2) is described. The TiB2 was deposited on mullite, alumina, and thermally oxidized silicon by a hydrogen reduction of titanium tetrachloride (TiCl4) and boron trichloride (BCl3). The process resulted in a reaction-limited regime, a heat-diffusion limited regime, and a mass-diffusion limited regime. The hydrogen concentration has a large influence on the growth rate, nucleation rate, and morphology of the coatings. The photothermal deposition process is photolytically enhanced by the 193 nm beam of an ArF excimer laser which dissociates TiCl4. However, the 193 nm beam enhances the nucleation rate and crystal-growth rate to a different extent. The nucleation rate increases with increasing atomic hydrogen concentration, as is illustrated by experiments performed with a separate atomic hydrogen source (i.e., hot filament-enhanced chemical vapor deposition).

Nucleation and growth of chemical vapor deposition TiN films on Si (100) as studied by total reflection x-ray fluorescence, atomic force microscopy, and Auger electron spectroscopy

We report for the first time the characteristics of the early growth of chemical vapor deposition (CVD) TiN films on Si (100) in the surface reaction limited regime, using total reflection x-ray fluorescence (TXRF), atomic force microscopy (AFM), and Auger electron spectroscopy (AES). Small amounts of CVD TiN from TiCl4 and NH3 reactants were deposited on Si (100) at 650 °C using a rapid thermal CVD system, at a total pressure of 155 mTorr. We examined these small amounts of CVD TiN by introducing TXRF as a tool for observing the initial stages of growth, and AFM for characterizing changes in the overall surface morphology. The nucleation and growth of the resulting CVD TiN films as determined by TXRF, AFM, and AES will be discussed.

Measurement of mass transport and reaction parameters in bulk solution using photobleaching. Reaction limited binding regime

Fluorescence recovery after photobleaching (FRAP) has been used previously to investigate the kinetics of binding to biological surfaces. The present study adapts and further develops this technique for the quantification of mass transport and reaction parameters in bulk media. The technique's ability to obtain the bulk diffusion coefficient, concentration of binding sites, and equilibrium binding constant for ligand/receptor interactions in the reaction limited binding regime is assessed using the B72.3/TAG-72 monoclonal antibody/tumor associated antigen interaction as a model in vitro system. Measurements were independently verified using fluorometry. The bulk diffusion coefficient, concentration of binding sites and equilibrium binding constant for the system investigated were 6.1 +/- 1.1 x 10(-7) cm2/s, 4.4 +/- 0.6 x 10(-7) M, and 2.5 +/- 1.6 x 10(7) M-1, respectively. Model robustness and the applicability of the technique for in vivo quantification of mass transport and reaction parameters are addressed. With a suitable animal model, it is believed that this technique is capable of quantifying mass transport and reaction parameters in vivo.

Selective low-pressure chemical vapor deposition of Si1−xGex alloys in a rapid thermal processor using dichlorosilane and germane

Low-pressure chemical vapor deposition of Si1−xGex alloys in a cold wall, lamp-heated rapid thermal processor was studied. Alloys were deposited using the reactive gases GeH4 and SiH2Cl2 in a hydrogen carrier gas. The depositions were performed at a total pressure of 2.5 Torr and at temperatures between 500 and 800 °C using GeH4:SiH2Cl2 ratios ranging from 0.025 to 1.00. Results showed that Si1−xGex alloys can be deposited selectively on silicon in SiO2. The selectivity is enhanced significantly by the addition of GeH4 in the gas stream. In this work, selective depositions were obtained when the GeH4:SiH2Cl2 gas flow ratio was greater than 0.2 regardless of the deposition temperature, corresponding to a Ge content of 20% or higher in the films as determined by Auger electron spectroscopy. An enhancement in the deposition rate was observed in agreement with earlier reports due to the addition of GeH4. The activation energy for deposition in the surface reaction limited regime varied from 20 to 30 kcal/mole with the gas flow ratios used in this study.