Showing papers on "Rate equation published in 1991"

Analysis of semiconductor microcavity lasers using rate equations

Abstract: The rate equations for a microcavity semiconductor laser are solved and the steady-state behavior of the laser and some of its dynamic characteristics are investigated. It is shown that by manipulating the mode density and the spontaneous decay rates of the cavity modes, the threshold gain can be decreased and the modulation speed can be improved. However, in order to fully exploit the possibilities which the modification of the spontaneous decay opens up, the active material volume in the cavity must be smaller than a certain value. Threshold current using different definitions, population inversion factor, L-I curves, linewidth, and modulation response are discussed. >

Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources

Abstract: A general theory of linewidth for single-frequency semiconductor lasers is presented. The effects of spatially distributed noise sources together with spatially varying carrier and photon densities and injection current are analyzed in a rigorous manner by solution of the scalar wave equation. A new rate equation for the electric field is derived, in which the longitudinal effects are represented in the form of the weight functions C/sub N/(z) and C/sub S/(z). These functions express the sensitivity of the (output) field to local changes in carrier and photon density at the position z. For Fabry-Perot laser's the z dependence of the C factors is shown to be negligible, in agreement with the fact that spatial hole burning is not considered to be important for Fabry-Perot lasers. For distributed-feedback (DFB) lasers, however, the z dependence is shown to be very significant. >

Theoretical approaches to reversible diffusion‐influenced reactions: Monomer–excimer kinetics

Abstract: Three formalisms that describe the influence of diffusion on the kinetics of the reversible reaction, A+B⇌AB, are discussed and compared. The simplest involves a modification of the irreversible rate equations of Smoluchowski theory; the second is based on a generalization of physically appealing convolution relations that hold rigorously for reversible reactions between isolated pairs, and the third can be obtained by using a superposition approximation to truncate the hierarchy of equations satisfied by the reactive reduced distribution functions. The various formalisms are developed to the point that their implementation requires knowledge only of the time‐dependent irreversible association rate coefficient and the microscopic dissociation rate constant. All these approaches give the correct equilibrium concentrations at infinite time, have the same short‐time behavior, reduce correctly when the dissociation rate is zero, and become equivalent in the reaction‐controlled limit. However, none of them pro...

Semiconductor laser stabilization by external optical feedback

Abstract: A report is presented on a general theory describing the effect of external optical feedback on the steady-state noise characteristics of single-mode semiconductor lasers. The theory is valid for arbitrarily strong feedback and arbitrary optical feedback configuration and spectrum. A generalized Langevin rate equation is derived. The equation is, in general, infinite order in d/dt constituting an infinite-order correction to the low-frequency weak-feedback analysis. The general formalism includes relaxation oscillations and permits analysis of the effect of feedback on the laser linewidth, frequency noise, relative intensity noise, and the relaxation oscillation sidebands in the field spectrum. The theory is applied to two important feedback configurations: the laser coupled to a single mirror and the laser coupled to a high-Q cavity. >

Kinetical Aspects of the LPCVD of Titanium Nitride from Titanium Tetrachloride and Ammonia

Abstract: was deposited from the reactant gases , , and , with Ar as a carrier gas. The depositions were carried out in a cold wall CVD reactor at a total deposition pressure of 20.0 Pa (150 mtorr). The deposition rate and film properties were studied as a function of the deposition temperature and the partial pressure of the reactant gases. A rate equation for has been determined No influence of on the deposition rate and film composition was detected. Even without , deposition occurred in a gas mixture of , , and Ar. This suggests that does not play a role in the deposition reaction mechanism. The reaction order for and can only be explained qualitatively because of the existence of a deposition reaction and a parallel complex forming reaction in the gas phase. These two reactions take place simultaneously and competitively. The complex forming reactions in the gas phase influence the reactant partial pressures. Until now this influence has not been known quantitatively. Therefore, a detailed reaction mechanism of the deposition could not be extracted from our results. An activation energy of 61 kJ/mol has been determined. At increasing deposition temperature and at higher values of the ratio, lower values of electrical resistivity were observed. This resistivity variation is found to be caused by the impurity content (Cl and O) in the films, which varies with the process parameters.

rf glow discharge of SiF4‐H2 mixtures: Diagnostics and modeling of the a‐Si plasma deposition process

Abstract: Hydrogenated and fluorinated amorphous silicon films (a‐Si:H,F) have been deposited in a parallel‐plate rf plasma reactor fed with SiF4‐H2 mixtures. The plasma phase characterization has been performed by optical emission spectroscopy for the analysis of the emitting species (SiF*3,SiF*2,SiF*,H*, and H*2), mass spectrometry for the analysis of stable species, and Langmuir electrical probes for the evaluation of electron density (ne) and temperature (kTe). The deposition rate (rD) has been monitored by laser interferometer. The effect of the rf power, gas composition, total pressure, and dopant (B2H6, PH3) addition on the plasma phase composition and on the film growth rate has been studied. The data have been discussed on the basis of a chemical model where the chemisorption of SiF4 and SiF2 and the subsequent interaction with H atoms are the determinant steps also from the kinetic point of view. It has been found that the rate equation rD ∝ [H][SiF2] is able to fit the experimental results.

Modulation of phased-array semiconductor lasers at K-band frequencies

Abstract: The possibility of phase locking the lateral modes of a multiple-emitter semiconductor-laser phased array by modulating at the beat frequency of the lateral modes is presented. A rate equation model of a twin-emitter array is introduced and used to find the response of the optical output of the array to small sinusoidal modulations of the injection current. In-phase modulation of the emitters is found to be ineffectual, while modulating the emitters 180 degrees out of phase extended the responsivity well beyond the relaxation oscillation frequency to approximately the frequency associated with the rate of photon exchange between the coupled emitters. The rate equations were numerically integrated for several cases of interest to evaluate the stabilizing effect of strong modulation (entrainment). >

Rate equations for the carbon-oxygen reaction: an evaluation of the Langmuir adsorption isotherm at atmospheric pressure

Abstract: The Langmuir adsorption isother has been tested for application to the carbon-oxygen reaction at atmospheric pressure against unique experimental data reported by Davis and Hottel in 1934; and values of activation energy and frequency factor have been obtained for both the adsorption and the desorption steps with E values of 10.04 and 32.95 kcal/mol, respectively. These values are within the expected ranges for the two steps. The evaluation required the solution of the four one-dimensional simultaneous diffusion equations for O 2 , N 2 , CO 2 with Stefan flow, to test the accuracy of the diffusion theory by comparision with experiment. Neglect of the Stefan flow is shown to result in a systematic 12% error in calculation. For the solutions to be tractable in conjunction with the reactivity equations they required linearization. The linearized equations were used to calculate the oxygen concentration at the surface of the reacting carbon samples, and the data obtained were used to validate the functional form of the Langmuir rate equation. The results confirmed in an unambiguous way that the reaction was two step, with one step, identified as adsorption, being first order with respect to oxygen concentration, and the other step, for desorption, being zero order

Diagnostics and modeling of RF discharge dissociation in N/sub 2/O

Abstract: Measurements were made of the RF discharge dissociation of N/sub 2/O in a parallel-plate reactor by downstream mass spectrometry using a wide range of gas flows and powers at 10 kHz and 13.56 MHz. The results show that the mass 44 signal (N/sub 2/O/sup +/), which is a measure of the amount of undissociated N/sub 2/O, is a function of the discharge input energy per N/sub 2/O molecule (eV/N/sub 2/O). A plug flow, rate equation model of the discharge was used to predict the experimental dissociation rates. A DC Monte Carlo simulation was used to calculate rate coefficients for electron-impact neutral dissociation, ionization, and dissociative ionization. The rate equation model also includes reactions among the dissociation products of N/sub 2/O and species which are synthesized in the discharge, as well as neural and electron-ion recombination at the electrodes. The model predictions identify the major reaction pathways and the sensitivities of the results to the rate coefficient values used. >

A Mixed Interface Reaction/Diffusion Control Model for Oxidation of Si3 N 4

Abstract: The observed parabolic rate constants for the oxidation of at 1100 to 1400°C are 10 to 1000 times lower than those for the oxidation of silicon. Calculations are performed to determine parabolic rate constants from the existing oxygen tracer diffusivity values and from the diffusivity and solubility values of the dissolved diatomic oxygen. These calculations suggest that the rate of oxidation is not controlled by oxygen diffusion. Instead, it is controlled by a mixed process involving reaction at the interface and nitrogen diffusion through . A detailed analysis indicates that the rate law can be nearly parabolic for this mixed control process.

Influence of coexisting analytes in atmospheric pressure ionization mass spectrometry

Abstract: A simple rate equation analysis was used to model the ion-molecule chemistry occurring in the ionization source. The experimental results obtained for the detection of dimethyl methyl phosphonate, in the presence of varying concentrations of diisopropyl methyl phosphonate, agree with the rate equation analysis

Thermochemical Kinetics of a Proposed Mechanism for Diamond Growth from Acetylene

Abstract: We have performed a thermochemical and kinetic analysis of a previously proposed mechanism for diamond growth from C2H2. We estimate enthalpy and entropy changes for each step in the mechanism using the MM3 molecular mechanics code. Rate constants are estimated based on the assumption that reaction cross sections measured for analogous gas phase reactions may be applied to gas-surface reactions. Gas phase species concentrations are taken from measurements and modeling. The rate equations are then integrated. According to the analysis, each step in the proposed mechanism is exothermic (ΔH<0), but the reduction in entropy is so great that this mechanism does not produce diamond at a significant rate. This result does not rule out the possibility of growth from C2H2 by some different reaction mechanism.

A theoretical model for intense microwave pulse propagation in an air breakdown environment

Abstract: A theoretical model based on two coupled partial differential equations is established to describe the propagation of an intense microwave pulse in air breakdown environment. One is derived from the Poynting theorem, and the other one is the rate equation of electron density. A semiempirical formula of the ionization frequency is adopted for this model. A transformation of these two equations to local time frame of reference is introduced so that they can be solved numerically with considerably reduced computation time.

A rate equation for atomic ordering in mean field theory: I. Uniform case

Abstract: A critical discussion is given of the kinetics of (dis)ordering in a simple system of Ising spins with respectively Glauber and Kawasaki types of stochastic processes. The question addressed is whether the rate is governed by the same free energy excess (in equilibrium) as assumed in frequently used rate equations. The authors focus for simplicity on uniform systems with long range interactions whose equilibria are well described by mean field theory. Such situations are realized in systems with effective spin interactions mediated by strain. They find that the rate equation can be expressed in terms of a Glauber or Kawasaki potential. Their analytical forms are found to be quite different from each other and also different from the equilibrium mean field energy. However, the predicted rate equations are not substantially different for most temperatures of interest.

Reaction Kinetics in Restricted Spaces

Abstract: (Received 2 Auglrrt 1990) Abstract. Reactions in restricted spaces rarely get stirred vigorously by convection and are thus controlled by diffusion. Furthermore, the compactness of the Brownian motion leads to both anomalous diffusion and anomalous reaction kinetics. Elementary binary reactions of the type A + A + Products, A + B + Products, and A + C + C + Products are discussed theoretically for both batch and steady-state conditions. The anomalous reaction orders and time exponents (for the rate coefficients) are discussed for various situations. Global and local rate laws are related to particle distribution functions. Only Poissonian distributions guarantee the classical rate laws. Reactant self-organization leads to interesting new phenomena. These are demonstrated by theory, simulations, and experiments. The correlation length of rcactant production affects the self- ordering length scale. These effects are demonstrated experimentally, including the stability of reactant segregation observed in chemical reactions in one-dimen- sional spaces, e.g., capillaries and microcapillaries. The gap between the reactant A (cation) and B (anion) actually increases in time and extends over millimeters. Excellent agreement is found among theory, simulation, and experiment for the various scaling exponents.

Non-equilibrium effects on forward and reverse rate coefficients of reversible reactions in solutions: A stochastic approach

Abstract: The kinetics of irreversible and reversible reactions in solution is studied by applying a stochastic approach in the sense of Teramoto and Shigesada. For sufficiently low reactant concentrations the random processes attributed to the reactions reduce to Markovian ones, i.e. rate equations follow as deterministic approximation form a master equation and rate coefficients from its transition rates. The forward and reverse rate constants for the reactions A+B⇌C, A+B⇌C+B and A+B⇌C+D derived in this way have the involved structure well-known from other papers. Our treatment does not confirm the objections to former results recently rised by Lee and Karplus [1].

Reversible trapping on a cubic lattice: Comparison of theory and simulations

Abstract: Simulations of the kinetics of the reversible diffusion-influenced reaction A+B⇄AB on a cubic lattice, with initial conditions [A] = [B], [AB]=0, are compared with the predictions of two approximate theoretical formalisms. The first involves a simple rate equation with rate coefficients that are proportional to the time-dependent rate coefficient for an irreversible reaction. The second, which is based on a superposition approximation, contains a rate coefficient that explicitly depends on the bulk concentrations. Both reduce to the Smoluchowski approach in the irreversible limit. The results obtained using the modified rate equation formalism are exact at short times, but tend to approach equilibrium too rapidly. The predictions of the computationally more demanding superposition formalism agree remarkably well with the simulations for all times for the range of parameters examined.

Exchange concept in water-in-oil microemulsions: consequences for ‘slow’ chemical reactions

Abstract: Water-in-oil (w/o) microemulsions are thermodynamically stable dispersions of water droplets in an oil-continuous medium. Water and solutes exchange between droplets in a random process whereby pairs of droplets fuse transiently and then separate again. An equation is derived for the rate of an irreversible chemical reaction between two water-soluble reactants A and B, initially isolated in separate droplets, by applying the steady-state approximation (i.e. by assuming that the concentration of intermediate droplet species containing both A and B, generated in the fusion process, is constant under initial rate conditions). The equation shows how the rate of the chemical reaction can be expected to depend on the rate of the fusion process and the size and concentration of the droplets. The theoretical basis for the utilization of very fast chemical reactions in the measurement of droplet fusion rates, exploited in earlier published studies, is discussed in the context of the rate equation. The analysis is extended to the case of chemical reactions involving a reversible step (i.e. of the type A + B ⇄ C → P). The potential for the additional perturbation of reaction rates arising from the influence of properties of the microemulsion aqueous environment not associated with dynamics of droplet interactions is emphasized. With the emphasis on ‘slow’ chemical reactions, a revision of nomenclature is recommended in order to distinguish the kinetics of the fusion process itself from those of exchange, which should be regarded as a multiple-step process incorporating both droplet fusion and solute-transfer events.

Relation between the thermodynamics and kinetics of a complex reaction system at constant temperature and pressure

Abstract: For complex systems, it is convenient to express both the thermodynamic condition for chemical equilibrium and the rate equation in matrix notation. Chemical equilibrium calculations at constant temperature and pressure are generally made on the assumption that the only constraints on the minimization of the Gibbs energy are the element balances. This can be accomplished in terms of chemical reactions (the so-called stoichiometric formulation of the problem) by the use of any independent set of R reactions that can provide for all possible compositions permitted by the conservation of elements. The rate equation for a complex system is written in terms of a mechanism that may involve a larger or smaller number of reactions. An equilibrium composition can be calculated from the rate equation by integration to infinite time, but this composition may or may not agree with that calculated from thermodynamics on the assumption that the only constraints are the element balances. If the mechanism includes R independent reactions, the equilibrium compositions calculated in these two ways should agree if the forward and reverse rate constants for each step are in agreement with the equilibrium constants for those steps. If the mechanism involves fewer than R independent reactions, amore » different equilibrium composition will be calculated from the rate equation because constraints are implicitly involved in the mechanism. This is illustrated for a system involving polycyclic aromatic hydrocarbons.« less

On the Presence of Limit-Cycles in a Model Exothermic Chemical Reaction: Sal'nikov's Oscillator with Two Temperature-Dependent Reaction Rates

Abstract: This paper investigates a model chemical reaction in which a single substance undergoes a two-stage process of decay, first producing an intermediate species and finally giving a product chemical. Each of the two stages involves only simple first-order reaction kinetics, but the governing rate parameter for each of the two reactions is temperature dependent. The reaction vessel is assumed to be well stirred, and the rate of each reaction is governed by Arrhenius kinetics, although with a different activation energy for each process. The mathematical behaviour of the system is therefore described by a coupled system of two highly nonlinear ordinary differential equations for the concentration of the intermediate species and the temperature, arising from the rate equation and the energy conservation equation. This simple model is capable of predicting oscillatory behaviour in the concentration of the intermediate chemical and in the temperature. We present the Hopf condition for the emergence of these limit cycles from a homogeneous steady state, and then continue these solutions numerically into regions of the parameter space in which oscillations of very large amplitude can occur. The presence of multiple limit cycles is detected and discussed. An extension of Bendixson’s criterion is used to show that oscillatory behaviour is only possible in a certain confined region of the parameter space.

Time resolved study of the bulk plasma of a 13.6 MHz discharge in argon

Abstract: Time dependent population densities of excited states of Ar and Ar+ have been measured by time and space resolved optical emission spectroscopy in an asymmetric parallel plate plasma reactor in Ar at an excitation frequency of 13.6 MHz. Time averaged ion densities in the plasma have been determined by a floating double probe. At pressures above 20 Pa and at distances from the powered electrode beyond the emission intensity maximum the population densities are modulated with essentially the fundamental RF frequency. In order to explain this experimental result the bulk plasma was modelled under the assumption that the local electric field is the sum of a time dependent part oscillating with the fundamental frequency and a constant part of similar size. For the calculations, the electron Boltzmann equation and a set of rate equations were simultaneously solved in a fully time dependent way. The observed modulations of the emission intensity of the spectral line of Arl at 516.23 nm and of the calculated level population of the emitting 6d(1/2)1 state agree within the experimental errors.

Kinetic Rate Laws as Derived from Order Parameter Theory V: Computer Simulations of Ordering Processes Using a Soft Ising Model

Abstract: The kinetic and equilibrium behaviour of order/disorder systems and related processes are simulated using an Ising spin model in which the coupling between spins occurs via local strain in a harmonic lattice. The equilibrium states are found to be well described by a mean field Landau-type Gibbs free energy. The ordering kinetics laws follow the rate law: $$\frac{{dQ}}{{dt}} = \frac{{ - 1}}{\tau }\frac{{\partial G}}{{\partial Q}}$$ which is the limit for a non-conserved order parameter of the general rate law discussed in this series of papers. The excess Gibbs free energy G is determined by the equilibrium states and differs substantially from that of the classical Ising nearest neighbour model. The degree of order remains rather uniform over the sample due to the long range interactions and the kinetics follow the predictions of a Landau-type mean field theory to a very good approximation. This is in strong contrast to the kinetics of a nearest neighbour Ising model.

A statistically based model of electron‐beam exposed, chemically amplified negative resist

Abstract: A methodology based on statistically designed factorial experiments has been developed to model the dissolution rate of an electron‐beam exposed chemically amplified resist, Shipley SAL‐601‐ER7, for different post‐exposure bake (PEB) and developer concentrations. A statistical experiment of dissolution rate measurements were performed in the Perkin‐Elmer development rate monitor using a central composite design with PEB temperature, PEB time, and developer concentration as the factors. These measurements were combined with Monte Carlo simulation of electron energy deposition to generate dissolution rate data as a function of absorbed energy. These data were then fitted to a semiempirical rate equation using nonlinear regression. Statistical analysis of the effects of the three processing factors on the parameters of the rate functions yielded empirical models relating the parameters to the factors. As a result, a rate equation can be determined for any processing conditions, and thus a complete model is o...