Kinetic Rate Equation Model Research Articles

Secondary Nucleation by Interparticle Energies. III. Nucleation Rate Model.

A nucleation rate model for describing the kinetics of secondary nucleation caused by interparticle energies (SNIPEs) is derived theoretically, verified numerically, and validated experimentally. The theoretical derivation reveals that the SNIPE mechanism can be viewed as enhanced primary nucleation, i.e., primary nucleation with a lower thermodynamic energy barrier (for nucleation) and a smaller critical nucleus size, both caused by the interparticle interactions and the associated energy between the surface of a seed crystal and a molecular cluster in solution, as shown in part I of this series. In the case of a sufficiently agitated suspension, the model depends on four parameters: two reflecting primary nucleation kinetics and the other two accounting for the intensity and effective spatial range of the interparticle interactions. As a numerical verification of the model, we show that the nucleation kinetics described by the SNIPE rate model is in quantitative agreement with those given by the kinetic rate equation model developed in part II of this series. A sensitivity analysis of the SNIPE rate model is conducted to present the effect of key model parameters on the nucleation kinetics. Moreover, the SNIPE rate model is validated by fitting the model to the time-resolved data of secondary nucleation experiments as well as to two other, well-known secondary nucleation rate models. Importantly, all of the estimated parameter values for the SNIPE model were consistent with the theoretical estimates, while some of the estimated parameter values for one of the well-known secondary nucleation models deviated from the corresponding theoretical values significantly.

Decay of excess carriers in a two-defect model semiconductor: A time-resolved photoluminescence study

The study of charge carrier decay dynamics in semiconductors plays an important role in determining various device properties, for example, the bulk minority carrier lifetime and the potential drop at interfaces. Time-resolved photoluminescence spectroscopy is an efficient technique to extract carrier decay time constants, with the simplest case being that of a mono-exponential decay controlled by a single type of recombination center. However, many modern materials often display multi-exponential decay behavior influenced by a broad distribution of defects. Utilizing CuIn1-xGaxSe2 as a reference system, we present a comprehensive and systematic theoretical study of the decay kinetics in a non-interacting two-defect-level system by using both the classical analytical approach developed by Hornbeck and Haynes and a homogeneous kinetic rate equation model solved numerically. We show how the trapping level's carrier capture cross-section symmetry factor and the initial injection level controls the parameter range for the agreement between these two approaches, analyzing the validity of multi-exponential fits for extracting the carrier lifetime. Further, we analytically and numerically demonstrate that information on the carrier recombination lifetime is contained in the fast time constant, whereas the agreement of the subsequent slow time constant with the analytical model depends on the energetic position of the defect level within the bandgap.

Secondary Nucleation by Interparticle Energies. II. Kinetics.

This work presents a mathematical model that describes growth, homogeneous nucleation, and secondary nucleation that is caused by interparticle interactions between seed crystals and molecular clusters in suspension. The model is developed by incorporating the role of interparticle energies into a kinetic rate equation model, which yields the time evolution of nucleus and seed crystal populations, as in a population balance equation model, and additionally that of subcritical molecular clusters, thus revealing an important role of each population in crystallization. Seeded batch crystallization at a constant temperature has been simulated to demonstrate that the interparticle interactions increase the concentration of the critical clusters by several orders of magnitude, thus causing secondary nucleation. This explains how secondary nucleation can occur at a low supersaturation that is insufficient to trigger primary nucleation. Moreover, a sensitivity analysis has shown that the intensity of the interparticle energies has a major effect on secondary nucleation, while its effective distance has a minor effect. Finally, the simulation results are qualitatively compared with experimental observations in the literature, thus showing that the model can identify operating conditions at which primary or secondary nucleation is more prone to occur, which can be used as a useful tool for process design.

Modified kinetic rate equation model for cooling crystallization

The kinetic rate equation (KRE) model, unlike the population balance equation model, can describe growth, nucleation, and even Ostwald ripening simultaneously. However, the KRE model cannot be applied in cooling crystallization systems. In this work, we propose a modified KRE model to describe cooling crystallization. The modified KRE model can successfully describe crystal growth and nucleation in cooling crystallization systems. In addition, the metastable zone width was simulated using the modified KRE model and compared with the experimental data in references. The results revealed that the modified KRE model could express the effect of overheating prior to cooling on the metastable zone width. As the extent of overheating increases, the metastable zone width becomes wider, which phenomenon can be clearly simulated by the modified KRE model. This modeling capability is attributed to the behavior of particle clusters that are sized less than the size of sub-nuclei. Because the population balance equation model cannot describe the metastable zone width, the modified KRE model has certain competitive advantages in its application to various crystallization systems.

Strongly temperature-dependent recombination kinetics of a negatively charged exciton in asymmetric quantum dots at 1.55 µm

We report on strongly temperature-dependent kinetics of negatively charged carrier complexes in asymmetric InAs/AlGaInAs/InP quantum dots (dashes) emitting at telecom wavelengths. The structures are highly elongated and of large volume, which results in atypical carrier confinement characteristics with s-p shell energy splittings far below the optical phonon energy, which strongly affects the phonon-assisted relaxation. Probing the emission kinetics with time-resolved microphotoluminescence from a single dot, we observe a strongly non-monotonic temperature dependence of the charged exciton lifetime. Using a kinetic rate-equation model, we find that a relaxation side-path through the excited charged exciton triplet states may lead to such behavior. This, however, involves efficient singlet-triplet relaxation via the electron spin-flip. Thus, we interpret the results as an indirect observation of strongly enhanced electron spin relaxation without a magnetic field, possibly resulting from atypical confinement characteristics.

Mechanically Induced Homochirality in Nucleated Enantioselective Polymerization.

Understanding how biological homochirality may have emerged during chemical evolution remains a challenge for origin of life research. In keeping with this goal, we introduce and solve numerically a kinetic rate equation model of nucleated cooperative enantioselective polymerization in closed systems. The microreversible scheme includes (i) solution-phase racemization of the monomers, (ii) linear chain growth by stepwise monomer attachment, in both nucleation and elongation phases, and (iii) annealing or fusion of homochiral chains. Mechanically induced breakage of the longest chains maintains the system out of equilibrium and drives a breakage-fusion recycling mechanism. Spontaneous mirror symmetry breaking can be achieved starting from small initial enantiomeric excesses due to the intrinsic statistical fluctuations about the idealized racemic composition. The subsequent chiral amplification confirms the model's capacity for absolute asymmetric synthesis, without chiral cross-inhibition and without explicit autocatalysis.

Cathodoluminescence spectrum imaging analysis of CdTe thin-film bevels

We conducted T= 6 K cathodoluminescence (CL) spectrum imaging with a nanoscale electron beam on beveled surfaces of CdTe thin films at the critical stages of standard CdTe solar cell fabrication. We find that the through-thickness CL total intensity profiles are consistent with a reduction in grain-boundary recombination due to the CdCl2 treatment. The color-coded CL maps of the near-band-edge transitions indicate significant variations in the defect recombination activity at the micron and sub-micron scales within grains, from grain to grain, throughout the film depth, and between films with different processing histories. We estimated the grain-interior sulfur-alloying fraction in the interdiffused CdTe/CdS region of the CdCl2-treated films from a sample of 35 grains and found that it is not strongly correlated with CL intensity. A kinetic rate-equation model was used to simulate grain-boundary (GB) and grain-interior CL spectra. Simulations indicate that the large reduction in the exciton band intensity and relatively small decrease in the lower-energy band intensity at CdTe GBs or dislocations can be explained by an enhanced electron-hole non-radiative recombination rate at the deep GB or dislocation defects. Simulations also show that higher GB concentrations of donors and/or acceptors can increase the lower-energy band intensity, while slightly decreasing the exciton band intensity.

Towards biochemical filters with a sigmoidal response to pH changes: buffered biocatalytic signal transduction

We realize a biochemical filtering process by introducing a buffer in a biocatalytic signal-transduction logic system based on the function of an enzyme, esterase. The input, ethyl butyrate, is converted into butyric acid--the output signal, which in turn is measured by the drop in the pH value. The developed approach offers a versatile "network element" for increasing the complexity of biochemical information processing systems. Evaluation of an optimal regime for quality filtering is accomplished in the framework of a kinetic rate-equation model.

Non-monotonous dependences of thin film surface roughness on substrate temperature and deposited atom flux

Experimentally observed non-monotonous dependences of thin films surface roughness on substrate temperature and flux of deposited atoms are analysed by kinetic rate equation model. The modelling results show good qualitative agreement with the experiment and explain an unusual phenomenon of surface roughness. It is shown that non-monotonous dependence of surface roughness on substrate temperature and deposition flux is determined by the size of islands and diffusivity of atoms on the surface. From present analysis it follows that the formation mechanisms of non-monotonous dependences of the surface roughness on temperature and deposition flux are of different origin. The mechanisms are qualitatively analysed in the presentpaper. Keywords: surface roughness, island films deposition, surface diffusion, kinetic modelling

Mathematical Modeling of in vitro Enzymatic Production of 2-Keto- L-gulonic Acid Using NAD(H) or NADP(H) as Cofactors

A 2-Keto- L-gulonic acid (2-KLG) production process using stationary Pantoea citrea cells and a Corynebacterium 2,5-diketo- D-gluconic acid (2,5-DKG) reductase enzyme has been developed which may represent an improved method of vitamin C biosynthesis. Experimental data was collected using the F22Y/A272G 2,5-DKG reductase mutant and NADP(H) as a cofactor. An extensive kinetic analysis was performed and a kinetic rate equation model for this process was developed. A recent protein engineering effort has resulted in several 2,5-DKG reductase mutants exhibiting improved activity with NADH as a cofactor. The use of NAD(H) in the bioreactor may be preferable due to its increased stability and lower cost. The kinetic parameters in the rate equation model have been replaced in order to predict 2-KLG production with NAD(H) as a cofactor. The model was also extended to predict 2-KLG production in the presence of a range of combined cofactor concentrations. This analysis suggests that the use of the F22Y/K232G/R238H/A272G 2,5-DKG reductase mutant with NAD(H) combined with a small amount of NADP(H) could provide a significant cost benefit for in vitro enzymatic 2-KLG production.

Rate-equation modeling of single- and multiple-quantum vibrational energy transfer of OH (A 2 Σ + , v ′ =0 to 3)

A computer code based on a kinetic rate-equation model for describing the collisional dynamics of OH (A 2Σ+) in laser-induced fluorescence experiments was developed. In this work, the capabilities of the simulation code are extended to include the vibrational states up to the OH (A 2Σ+, v′=3) level. The calculation of quenching, rotational and vibrational relaxation rate coefficients for different collider species is discussed. Problems that arise for the description of vibrational relaxation include the branching ratio between single- and multiple-quantum steps and the form of the nascent rotational distribution after a vibrational relaxation step. Experimental spectra recorded under a variety of conditions are simulated using a consistent set of model assumptions. The calculations must include vibrational relaxation steps up to Δv=3 to account for the experimental intensity distributions. Effects due to polarized laser excitation become more important for vibrational states with v′>1. Areas for future work are identified, including determination of experimental rate coefficients for state-changing and depolarizing collisions in the upper vibrational levels.

A study of the kinetic rate equation model for simulations of molecular beam epitaxy crystal growth: temperature dependence of surface kinetic processes

In this paper, we have studied the rate equation model of Kariotis and Lagally that can be used for a computer simulation of molecular beam epitaxial (MBE) growth. The advantage of such a model is the high speed of calculations so that results can be obtained in a very short time. The model is based on a solution of kinetic rate equations that govern the time dependence of the concentration of islands of varying sizes and differing heights on the top of the surface during MBE growth. One of the results of this model is an expression for the time dependence of the coverage of each layer . Within the present approach the MBE growth mode is evaluated by a calculation of the RHEED (reflection high-energy electron diffraction) intensity I(t). Similarly a calculation of the interface width IW(t), which determines the roughness of the growing surface, serves to confirm the growth mode of the MBE structure. In original model of Kariotis and Lagally the MBE growth has been affected by an appropriate choice of a set of the RE input parameters describing the probability of various kinetic processes appearing on the surface during the growth. In order to make the model applicable to simulations of MBE growth at `real' growth parameters (substrate temperature and deposition), we have included the substrate temperature dependence of all surface kinetic processes in the model. In this manner, and including processes described in the text, we have created a qualitatively new model that describes the MBE growth more accurately. This model has been used to simulate the MBE growth of GaAs at different substrate temperatures. The results have been compared with MC simulations and give an agreement in coverage profile

A study of a kinetic rate equation model for simulations of MBE crystal growth: a comparison with Monte Carlo simulations

We present the simulations of molecular beam epitaxy (MBE) growth using a rate equation (RE) model and its comparison with Monte-Carlo (MC) simulations. The advantage of the RE model is the higher speed of calculations, so a much shorter time is required for obtaining results. The RE model is described by a set of differential equations that calculate at each time interval the change of the N kj numbers of atoms and islands of each k size in each jth layer. This change is due to kinetic processes occurring on the surface during the growth. In the original model (R. Kariotis and H.G. Lagally, Surf. Sci., 216 (1989) 557) the probabilities of these processes were described by parameters (input parameters for equations) and the simulations of MBE growth were realized by an appropriate choice of them. To make this model applicable to real simulations, we have included the substrate-temperature dependence of all input parameters using an Arrhenius form. This form is used in MC simulations to calculate a migration of atoms on the surface with substrate-temperature dependence. Since the RE model is described by a set of differential equations it was important to first find the allowed temperature range for simulations. This range includes the substrate temperature for the 3D growth mode (low temperatures) and also for the 2D growth mode (epitaxial temperatures). Using an Arrhenius form for temperature dependence of the parameters in the RE model we were able to compare the obtained results with MC calculations. We have made MC simulations (Š. Németh, R. Harman and M. Veselý, Correlation between the stochastic simulation of molecular beam epitaxy growth and experiment, 9th Int. Conf. of Thin Films, 6–10 September, 1993, Vienna, Austria) using the same input parameters ( T = 775 K, E n = 0.3 eV, E s = 1.45 eV). Since the RE model is strongly substrate-size dependent (D. Papajová, W.E. Hagston and P. Harrison, Appl. Phys. A, 59 (1994) 215–222; D. Papajová, Š. Németh, W.E. Hagston, H. Sitter and M. Veselý, J. Appl. Phys. A, submitted) we have found very good agreement in 2D growth for smaller substrate sizes S (in the RE model) only, when this dependence does not influence the results.

The simulation of single-crystal growth by molecular beam epitaxy using a kinetic rate-equation model

A computer simulation of Molecular Beam Epitaxial (MBE) growth under typical growth conditions is presented. The method is based on a solution of kinetic rate equations that govern the time dependence of the concentration of islands of varying sizes and differing heights on the top of the surface during the MBE growth. The time dependence for the coverage of each monolayer (θ) is calculated. The mode of the MBE growth is determined by a calculation of the RHEED (Reflection High-Energy Electron Diffraction) intensity. A calculation of the Exposed Coverage (EC) (which is defined as the number of surface atoms in each layer unscreened by other atoms directly above them) and the interface width (IW) (which determines the roughness of the growing surface) serves to confirm the growth mode of the MBE structure. The possible role of these type of calculations in determining the optimized growth conditions for the production of 2D growth in a general materials system is also described.

Directed picosecond excitation transport in purple photosynthetic bacteria

Picosecond spectrally resolved fluorescent measurements together with the coupled kinetic rate equation model simulations of the data have been employed to determine the rates and pathways of heterogeneous excitation transport in the membranes of photosynthetic bacteria Rhodobacter sphaeroides and Chromatium minutissimum. Excitation transport from bacteriochlorophyll molecules B800 to B850, belonging to the same pigment-pigment-protein complex B800–850, proceeds in 1–2 ps at room temperature and at 77 K. The intercomplex excitation transport from B850 to B875 takes about 10 ps for most excitations. In the minor part of B800–850 complexes, the excitation transport to B875 complex takes much longer, about 50 ps. The macroscopic rate constant of excitation trapping by open reaction centres is shown to be the same for all the bacteria studied, although the number of antenna molecules per reaction centre differs significantly. This seems to be an indication of the intrinsic homogeneity of the long-wavelength bacteriochlorophyll band, which facilitates an additional localization of excitations in the vicinity of the reaction centre and, due to the shortening of the trapping time, increases the overall quantum yield of photosynthesis.