Field Rate Equations Research Articles

Current modulation/optical injection and feedback for semiconductor laser diode based on optical field rate and intensity rate equation model

Current modulation/optical injection and feedback for semiconductor laser diode based on optical field rate and intensity rate equation model

Numerical simulation of multi-color laser generation in Tm-doped tellurite microsphere at 1.9, 1.5 and 2.3 microns

We present the first detailed theoretical analysis of multi-color continuous wave lasing in Tm-doped tellurite spherical microresonators with whispering gallery modes pumped at a wavelength of 792 nm. The numerical model is based on solving a system of equations for intracavity field amplitudes and rate equations using the parameters of Tm-doped tellurite glass measured in the previous experiments. All fundamental whispering gallery modes in the gain bands are taken into account. We demonstrate diagrams of generation regimes depending on Q-factors and pump power, which show a possibility of single-color lasing at a wavelength of ~1.9 μm, two-color lasing at wavelengths of ~1.9&1.5 μm and at ~1.9&2.3 μm, and three-color lasing at wavelengths of ~1.9&1.5&2.3 μm. Such microlasers can play a significant role in sensing applications.

Nonlinear properties of chloroindium phthalocyanines with nanosecond pulses

In this work we presented the study on nonlinear dynamics of tetra-α-(2-ethylbutoxy) chloroindium phthalocyanine (α-InPcCl) and tetra-β-(2-ethylbutoxy) chloroindium phthalocyanine (β-InPcCl). We replaced the electronic structure of molecules with the five-level model, and solved two-dimensional paraxial field equation and rate equations by Crank-Nicholson numerical method. The main passageway of photon absorption for optical limiting is the sequential (singlet-singlet) × (triplet-triplet) two-photon absorption. Both phthalocyanines showed distinctive optical limiting behaviors with a 532 nm–7 ns nanosecond laser. α-InPcCl shows better optical limiting effect, and we took it as an example to studied the optical dynamics in detail.

Improved optical limiting of triazine-linked mono-, bis- and tris-phthalocyanines

We studied the photophysical dynamics and optical limiting properties of a series of triazine-linked mono-, bis-, and tris-phthalocyanines with central metal In for 532nm-7ns laser pulses. We used Crank-Nicholson numerical method to solve two-dimensional paraxial field equation and rate equations. Phthalocyanines are simplified to a five-level model to simulate the optical dynamical processes. We found the sequential (singlet–singlet) × (triplet–triplet) two-photon absorption is the prominent optical limiting mechanism for long pulses. Our work indicates that triazine-linked mono-, bis-, and tris-phthalocyanines exhibit excellent optical limiting behaviors, and tris-phthalocyanine shows the best properties.

Mathematical theory of exchange-driven growth

Exchange-driven growth is a process in which pairs of clusters interact by exchanging single unit of mass at a time. The rate of exchange is given by an interaction kernel which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field rate equations of this process for the first time. We find two different classes of behavior depending on whether is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying . This result is optimal in the sense that we show for a large class of initial conditions and kernels satisfying the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for ( while existence is lost for ( In the intermediate regime we can only show local existence. We conjecture that the intermediate regime exhibits finite-time gelation in accordance with the heuristic results obtained for particular kernels.

Mechanistic Picture and Kinetic Analysis of Surface-Confined Ullmann Polymerization.

Surface-confined polymerization via Ullmann coupling is a promising route to create one- and two-dimensional covalent π-conjugated structures, including the bottom-up growth of graphene nanoribbons. Understanding the mechanism of the Ullmann reaction is necessary to provide a platform for rationally controlling the formation of these materials. We use fast X-ray photoelectron spectroscopy (XPS) in kinetic measurements of epitaxial surface polymerization of 1,4-dibromobenzene on Cu(110) and devise a kinetic model based on mean field rate equations, involving a transient state. This state is observed in the energy landscapes calculated by nudged elastic band (NEB) within density functional theory (DFT), which assumes as initial and final geometries of the organometallic and polymeric structures those observed by scanning tunneling microscopy (STM). The kinetic model accounts for all the salient features observed in the experimental curves extracted from the fast-XPS measurements and enables an enhanced understanding of the polymerization process, which is found to follow a nucleation-and-growth behavior preceded by the formation of a transient state.

Nonlinear dynamics and optical power limiting of nanoseconds pulses in naphthalocyanines and phthalocyanines with central metals gallium and indium

Abstract The optical dynamics of octa-(4-tert-butylphenoxy) substituted naphthalocyanines and phthalocyanines with different central metals (Ga, In) were investigated for 532 nm–7 ns laser pulses. In this work, we used numerical theoretical method with two-dimensional paraxial field equation and rate equations solving by Crank–Nicholson method. A five-level model was proposed to simulate the dynamical processes and the prominent optical limiting mechanism with long pulses is the sequential (singlet–singlet) × (triplet–triplet) two-photon absorption. The theoretical results indicate that both naphthalocyanines and phthalocyanines exhibit good optical limiting performances, which is attributed mainly to the reverse saturable absorption mechanism. The naphthalocyanines with larger difference of absorption cross sections and faster rate of the intersystem crossing transition between triplet and singlet states exhibit better optical limiting behaviour.

A simple model of burst nucleation.

We introduce a comprehensive quantitative treatment for burst nucleation (BN)-a kinetic pathway toward self-assembly or crystallization defined by an extended post-supersaturation induction period, followed by a burst of nucleation, and finally the growth of existing stable assemblages absent the formation of new ones-based on a hybrid mean field rate equation model incorporating thermodynamic treatment of the saturated solvent from classical nucleation theory. A key element is the inclusion of a concentration-dependent critical nucleus size, determined self-consistently along with the subcritical cluster population density. The model is applied to an example experimental study of crystallization in tetracene films prepared by organic vapor-liquid-solid deposition, where good agreement is observed with several aspects of the experiment using a single, physically well-defined adjustable parameter. The model predicts many important features of the experiment, and can be generalized to describe other self-organizing systems exhibiting BN kinetics.

Impact of soft impingement on the kinetics of diffusion-controlled growth of immiscible alloys

The effect of the soft impingement mechanism on the kinetics of diffusion-controlled phase transitions of binary alloys is investigated. Soft impingement arises from the interference of the diffusional fields and implies a different morphology of the product phase when compared to that attained in the case of genuine impingement. Mean field rate equations for nucleus radius and supersaturation are coupled with the mass balance equation in terms of the geometrical parameter (time dependent) which defines the soft impingement process. The kinetics is studied as a function of the initial supersaturation and it is compared with that attained in the case of genuine impingement. Solutions of the kinetics are obtained in the model case of time dependent supersaturation which scales according to Ham’s law and in the case of simultaneous nucleation. The phase transformation of immiscible alloys in one-dimensional system is also investigated. In this case and for point islands, the exact solution of the kinetics is obtained and allows one to gain an insight into the validity of the Kolmogorov–Johnson–Mehl–Avrami model and Avrami’s exponent for one-dimensional growths.

Island size distributions in submonolayer growth: Prediction by mean field theory with coverage dependent capture numbers

We show that mean field rate equations (MFRE) for submonolayer growth with hit and stick aggregation can successfully predict the island size distributions (ISDs) in the precoalescence regime if the full dependence of capture numbers on both the island size and the coverage is taken into account. This is demonstrated by comparison of the integrated MFRE with results from extensive kinetic Monte Carlo simulations. The attempt to use various simplified expressions for the capture numbers in the integration of the MFRE turns out to be insufficient to yield a good description of the ISD.

Thermoacoustic instability in a solid rocket motor: non-normality and nonlinear instabilities

An analytical framework is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into account the non-orthogonality of the eigenmodes of the unsteady coupled system. The coupled system comprises the dynamics of the acoustic field and the propellant burn rate. In general, thermoacoustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, the classical linear stability predicted from the eigenvalue analysis is valid in the asymptotic (large time) limit. However, the short-term dynamics can be completely different and a generalized stability theory is needed to predict the linear stability for all times. Non-normal systems show an initial transient growth for suitable initial perturbations even when the system is stable according to the classical linear stability theory. The terms contributing to the non-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in this analysis. Furthermore, the short-term dynamics are analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than usingad hocresponse functions which were used in earlier analyses. In this paper, a solid rocket motor with homogeneous propellant grain has been analysed. Modelling the evolution of the unsteady burn rate using a differential equation increases the degrees of freedom of the thermoacoustic system. Hence, a new generalized disturbance energy is defined which measures the growth and decay of the oscillations. This disturbance energy includes both acoustic energy and unsteady energy in the propellant and is used to quantify the transient growth in the system. Nonlinearities in the system are incorporated by including second-order acoustics and a physics-based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to ‘pulsed instability’. Pulsed instability is shown to occur with pressure coupling for burn rate response. Transient growth is shown to play an important role in pulsed instability.

Steady state energy distribution function of adatoms in reactive adsorption

The interplay between the energy distribution function of adatoms and the rate of diatom formation in catalytic reaction is studied by means of mean field rate equations. The recombination of adatoms is described as a multi-channel process where adatom desorption arises from several energy levels. It is shown that the distribution function can be computed, analytically, as a function of the ratio between recombination probability and rate constant for energy disposal to the solid. This parameter is the key quantity of the kinetics since it governs both reaction rate and the shape of the energy distribution function. It is found that, in order to obtain steady state conditions, the control parameter is constrained within a well defined interval of values which result lower than unity. It is shown that a kinetic transition takes place for the highest value of the parameter, which entails hyperthermal reaction rate.

A model kinetics for nucleation and diffusion-controlled growth of immiscible alloys

A system of mean field rate equations is employed for describing the kinetics of solid-solid phase separation within the immiscibility gap of binary alloys. The system allows us to study the time evolution of both supersaturation and diffusion length of the components in the metastable phase. It is shown that in the case of simultaneous nucleation the system of differential equations leads to a simple formula for the characteristic time of the transformation in terms of material parameters and initial supersaturation. The nucleation rate is computed on the basis of the classical nucleation theory and the alloy is assumed to behave as a regular solution. It turns out that for low values of the initial supersaturation the nucleation process can be considered as simultaneous. It is also found that thermally activated nucleation takes place for supersaturation values lower than about 0.21. The assumption of a concentration-independent diffusion coefficient and the effect of nucleus curvature on interface composition have been analyzed and discussed.

Impact of electron–hole pair excitation on the kinetics of exoergic processes at metal surfaces

The effect of energy disposal to metal surfaces, via electron–hole (e–h) pair excitation, on the rate of exoergic processes such as adsorption and adatom recombination, has been investigated. The modeling is based on the mean field rate equations and allows one to couple e–h pair excitation with either recombination or stimulated-desorption processes within a unique kinetic scheme. The energy distribution function of the metal electrons and the number of electrons available for detection as “chemicurrent” have been computed and compared to experimental results on chemicurrent yield for the H–Cu system. The kinetic model can also be applied to interpret experimental data on H(D)-adatom abstraction and recombination and on adsorption stimulated desorption of CO from metal surfaces. Rate coefficients of these desorption processes are proper for non-equilibrium energy distribution functions of the adlayer, which depend on the flux of adsorbing species and correlate to the surface electron density as modified by the adsorbate.

Kinetic phase diagram for island nucleation and growth during homoepitaxy

The impact of small-island dissociation and mobility on island nucleation and growth during vapor deposition of thin films is analyzed using mean field rate equations. The dominant island nucleation and growth mechanism is mapped onto a temperature/deposition-rate kinetic phase diagram, using Cu(100) homoepitaxy as an example. The methodology provides analytical expressions for the boundaries on the diagram and encourages a deeper understanding of the growth mechanisms. A kinetic Monte Carlo simulation incorporating the small-island dynamics is also presented and used to test the kinetic phase diagram, and satisfactory agreement is found throughout.

Unbounded autocatalytic growth on diffusive substrate: The extinction transition

The effect of diffusively correlated spatial fluctuations on the proliferation-extinction transition of autocatalytic agents is investigated numerically. Reactants adaptation to spatio-temporal active regions is shown to lead to proliferation even if the mean field rate equations predict extinction, in agreement with previous theoretical predictions. While in the proliferation phase the system admits a typical time scale that dictates the exponential growth, the extinction times distribution obeys a power law at the parameter region considered.

Multiscale modeling of island nucleation and growth duringCu(100)homoepitaxy

The long-time scale dynamics of small $\mathrm{Cu}∕\mathrm{Cu}(100)$ islands are studied. Atomistic simulations using embedded atom method (EAM) potentials and the dimer method saddle point searches provide pathways and their temperature-dependent rates to lattice-based kinetic Monte Carlo (KMC) simulations. The KMC utilizes translational symmetry to identify previously visited sites and re-use the atomistic rates. As a result very long time scales are accessible to the simulation which reveals the dissociation as well as the diffusion mechanisms of the small islands in an unbiased manner. Our results for island diffusion reproduce well the activation energies calculated in previous work, and provide in addition the associated frequency prefactors. The island dissociation pathways are rationalized in terms of previously anticipated mechanisms. We also utilize our results in mean field rate equations to predict ``kinetic phase diagrams'' for the critical island size as a function of temperature and vapor deposition rate during $\mathrm{Cu}(100)$ homoepitaxy. We predict that the higher critical island sizes $(i>2)$ should be observable at higher temperatures (above $\ensuremath{\sim}500\phantom{\rule{0.3em}{0ex}}\mathrm{K}$) at experimentally accessible deposition rates.

Mean field rate equation for diffusion-controlled growth in binary alloys

A system of rate equations is developed for modeling the kinetics of diffusional phase transitions in binary alloys. By employing a mean field assumption and the quasi static approximation a novel expression for the growth law of the nuclei is obtained in terms of supersaturation and diffusion length of component in the parent phase. The mean field rate equations are integrated for the model case of a transformation ruled by simultaneous nucleation and the behaviour of both Avrami’s exponent and rate constant investigated as a function of the initial value of the supersaturation. The expression of the activation energy, as extracted by the Arrhenius plot of the rate constant, is determined and is shown to be a function of the activation energies for nucleation and diffusion, and of the initial value of the supersaturation. The limiting case of infinite diffusion length is also analysed and discussed.

Epitaxial growth kinetics in the presence of an Ehrlich–Schwoebel barrier: comparative analysis of different models

In the last years, much theoretical effort has been devoted to the effects of the Ehrlich–Schwoebel (ES) step-edge barrier to interlayer diffusion on epitaxial growth. In a number of models, in a frame of the rate equation approach the expressions for critical island size for second-layer nucleation have been derived and various growth mode transitions, induced by the ES barrier, have been revealed, and corresponding ‘phase diagrams’ of the growth mode in a parameter space have been constructed. The application of these models to the experimental data has been shown to result in the reasonable estimates of the ES barrier. However, in just published papers by employing kinetic MC simulations and theory, based on a concept of the residence time of an adatom on top of an island and simple statistical arguments, the expressions for critical island size and nucleation rate differing from that of the rate equation approach, have been obtained and it has been remarked that the applicability of mean field rate equations to the confined geometry on top of a small island is not obvious. This article presents a comparative analysis and reconsideration of the rate equation-based models in view of these critical comments that shows that these models are valid at least in a range of weak ES barriers where the most interesting growth phenomena are developed.

Kinetics of Trapping Reactions with a Time Dependent Density of Traps

We study a trapping process where the traps ( particles B), besides being mobile, have a variable number. We analyze two cases related with the coupled reactions: A 1 B ! B, B 1 C ! C, and A 1 B ! B, B 1 C ! 0. It is shown that the time evolution of the traps strongly influences the kinetics of the trapping process, yielding qualitatively different behavior in both cases. The results of a model, adapted from one used before for trapping and annihilation in a one dimensional diffusion-limited system, have been compared with simulations yielding good qualitative agreement. [S0031-9007(97)02752-X] PACS numbers: 82.20.Hf, 05.60. + w The important role played by diffusion-controlled reactions in the most diverse branches of chemistry, physics, and biology has attracted the interest of researchers into the study of these problems during the last couple of decades [1]. This interest was motivated by the so-called “anomalous” kinetic laws that govern the evolution of these chemical reactions as in low dimensional systems (d # 2); they depart from the standard mean field rate equations [1,2]. In general, the kind of problems that have been studied include coalescence and annihilation reactions in one or two-species systems [1,2]. Such systems show a remarkable sensitivity in the kinetics of the recombination process and segregation to changes on initial conditions, presence of sources, disorder, external forces, etc. [1 ‐4]. Most of the recent literature is devoted to the analysis of these phenomena under the assumption that some kinds of rate equations are valid, considering the case of perfect reactions and, with lesser emphasis, in systems of partially absorbing media, which are of particular interest in many problems of attenuation in biological and physical problems [5 ‐ 8]. In this Letter we address the problem of a trapping reaction (symbolically written A 1 B ! B) in a one dimensional system of diffusing A particles and B traps, but in the case where the number of traps is time dependent. This dependence can arise because the traps participate in another reaction or because they are externally controlled. Such a situation, which has not been treated previously in the scientific literature, besides its interest in relation with several problems related to heterogeneous reactions and catalysis, shows some peculiarities that makes relevant its study on its own right. Here we will consider the following two related situations: (a) A 1 B ! B, B 1 C ! C (double trapping). (b) A 1 B ! B, B 1 C ! 0 (trapping with annihilated traps). In both cases it is clear that the second reaction will not be affected at all by the first one. This allows us to exploit known results for trapping and annihilation. In case (b), as usual, we restrict ourselves to equal initial densities of B and C particles in the annihilation reaction. In the present work, we show the results of Monte Carlo simulations made for both cases, and comparisons with the result of a mean field evaluation and with another theoretical model, which is a version of the Galanin model [8 ‐ 11], adapted to the present situation. First we present the mean field results in both situations. In case (a) the solution is given by