Constantinos Theodoropoulos

'Coarse' integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods

Abstract We present a time-stepper based approach to the ‘coarse’ integration and stability/bifurcation analysis of distributed reacting system models. The methods we discuss are applicable to systems for which the traditional modeling approach through macroscopic evolution equations (usually partial differential equations, PDEs) is not possible because the PDEs are not available in closed form. If an alternative, microscopic (e.g. Monte Carlo or Lattice Boltzmann) description of the physics is available, we illustrate how this microscopic simulator can be enabled (through a computational superstructure) to perform certain integration and numerical bifurcation analysis tasks directly at the coarse, systems level. This approach, when successful, can circumvent the derivation of accurate, closed form, macroscopic PDE descriptions of the system. The direct ‘systems level’ analysis of microscopic process models, facilitated through such numerical ‘enabling technologies’, may, if practical, advance our understanding and use of nonequilibrium systems.

Order reduction for nonlinear dynamic models of distributed reacting systems

Abstract Detailed first-principles models of transport and reaction (based on partial differential equations) lead, after discretization, to dynamical systems of very high order. Systematic methodologies for model order reduction are vital in exploiting such fundamental models in the analysis, design and real-time control of distributed reacting systems. We briefly review some approaches to model order reduction we have successfully used in recent years, and illustrate their capabilities through (a) the design of an observer and stabilizing controller of a reaction-diffusion problem and (b) two-dimensional simulations of the transient behavior of a horizontal MOVPE reactor.

Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

Model reduction and coarse-graining are important in many areas of science and engineering. How does a system with many degrees of freedom become one with fewer? How can a reversible micro-description be adapted to the dissipative macroscopic model? These crucial questions, as well as many other related problems, are discussed in this book. Specific areas of study include dynamical systems, non-equilibrium statistical mechanics, kinetic theory, hydrodynamics and mechanics of continuous media, (bio)chemical kinetics, nonlinear dynamics, nonlinear control, nonlinear estimation, and particulate systems from various branches of engineering. The generic nature and the power of the pertinent conceptual, analytical and computational frameworks helps eliminate some of the traditional language barriers, which often unnecessarily impede scientific progress and the interaction of researchers between disciplines such as physics, chemistry, biology, applied mathematics and engineering. All contributions are authored by experts, whose specialities span a wide range of fields within science and engineering

Design of gas inlets for the growth of gallium nitride by metalorganic vapor phase epitaxy

Abstract The problem of gas inlet design for metalorganic vapor phase epitaxy (MOVPE) of group III nitrides from metal alkyls and ammonia is addressed. The focus is on GaN growth from trimethyl-gallium and ammonia. In traditional reactors with well-mixed inlet streams, parasitic gas-phase reactions between the two precursors may lead to the formation of stable adducts that can condense on cool inlet walls, thus reducing the film growth efficiency. Such reactions may also lead to the formation of particulates through gas-phase condensation reactions (e.g. during AlN growth). A fundamentally-based model was developed to describe the MOVPE of GaN and was used to study the effect of inlet design and reactor operating conditions on film thickness uniformity in vertical stagnation-flow and rotating-disk reactors. The model includes a description of gas-phase kinetics and a simple gas-surface reaction mechanism. The kinetic model was coupled to a two-dimensional transport model describing flow, heat and mass transfer in a vertical MOVPE reactor. Predictions of growth rate compare well to experimental observations from a vertical rotating-disk reactor, without any adjustable parameters. The model was also used to study the distribution of gaseous species in the reactor and their role in film growth. Finite element simulations using a massively parallel computer code (MPSalsa) indicate that the species responsible for film growth are Ga-alkyls and not their adducts with ammonia. Sensitivity analysis was also performed to assess the relative importance of each reaction in determining the growth rate. The model was subsequently employed in the design of axisymmetric, multi-aperture gas inlets feeding precursors into the reactor in an alternating (not well-mixed) fashion. Simulations were performed to study the effect of key design parameters, such as inlet velocities, susceptor rotating speed, inlet to susceptor distance as well as the number and distribution of inlets, on GaN film growth rate and uniformity in industrial scale reactors. Optimal cases are presented that lead to uniform films over large-area substrates. An alternating precursor feed scheme based on concentric rings was found to lead to more uniform films as the number of inlets increased. The other important reactor parameters were inlet velocity, relative size of inlet “rings” and susceptor distance from the inlet.

Fundamental models of the metalorganic vapor-phase epitaxy of gallium nitride and their use in reactor design

A fundamental reaction-transport model describing the metalorganic vapor-phase epitaxy (MOVPE) of GaN from trimethyl-gallium (TMG) and ammonia has been developed. This model has been tested against experimental data from research-scale and industrial-scale reactors, A simplified version of the model that includes only transport phenomena and a unity sticking coefficient of the limiting film precursor (TMG) to the surface of the growing film was found to accurately capture observed film deposition variations in an early variant of the Thomas Swan close-coupled-showerhead 3 × 2 reactor. Modifications of the Thomas Swan reactor, in line with the findings suggested by this work, enabled state-of-the-art thickness uniformity to be achieved. The model has been used to develop performance diagrams for conceptual multi-aperture MOVPE reactors and for the Thomas Swan system. These performance diagrams identify regions of the parameter space of the reactor which correspond to minimal variations in film growth rate across large-area substrates. Published by Elsevier Science B.V.

Reaction kinetics and transport phenomena underlying the low-pressure metalorganic chemical vapor deposition of GaAs

Abstract A kinetic model describing the growth of GaAs films from triethyl-gallium (TEGa) and arsine by low-pressure metalogranic chemical vapor deposition (LP-MOCVD) has been developed. This precursor combination produces GaAs films with very low carbon contamination, when compared to films grown from trimethyl-gallium (TMGa) and arsine. The kinetic model includes both gas-phase and surface reactions based on reported decomposition mechanisms of the two precursors. Growth experiments were performed in a LP-MOCVD reactor operating at 3 Torr, a pressure that minimizes background levels of both ionized donors and acceptors in the film. The kinetic model was coupled to a transport model describing flow, heat and mass transfer in the experimental reactor. Finite element simulations were performed to estimate unknown rate parameters of surface growth reactions by fitting predicted to observed growth rates. The model predicts that the growth rate is limited by site-blocking effects due to slow desorption of adsorbed ethyl radicals at low temperatures and by the fast desorption of adsorbed gallium at high temperatures. The predicted Ga* coverage is high at intermediate temperatures suggesting the absence of a diffusion-limited growth regime, which is common in atmospheric pressure MOCVD. It appears that in LP-MOCVD, the temperature uniformity of the susceptor (and not gas-phase transport processes) controls the thickness uniformity of the film at operating conditions maximizing the growth rate. The proposed process model can evolve into a useful tool for reactor design and scale-up by minimizing the large number of experimental trial and error runs typically required to identify optimal operating conditions in MOCVD reactors.

Equation-free gaptooth-based controller design for distributed complex/multiscale processes

Abstract We present and illustrate a systematic computational methodology for the design of linear coarse-grained controllers for a class of spatially distributed processes. The approach targets systems described by micro- or mesoscopic evolution rules, for which coarse-grained, macroscopic evolution equations are not explicitly available. In particular, we exploit the smoothness in space of the process “coarse” variables (“observables”) to estimate the unknown macroscopic system dynamics. This is accomplished through appropriately initialized and connected ensembles of micro/mesoscopic simulations realizing a relatively small portion of the macroscopic spatial domain (the so-called gaptooth scheme). Our illustrative example consists of designing discrete-time, coarse linear controllers for a Lattice-Boltzmann model of a reaction-diffusion process (a kinetic-theory based realization of the FitzHugh-Nagumo equation in one spatial dimension).

An Input/Output Model Reduction-Based Optimization Scheme for Large-Scale Systems

A reduced model based optimization strategy is presented for the cases where input/output codes are the process simulators of choice, and thus system Jacobians and even system equations are not explicitly available to the user. The former is the case when commercial software packages or legacy codes are used to simulate a large-scale system and the latter when microscopic or multiscale simulators are employed. When such black-box dynamic simulators are used, we perform optimization by combining the recursive projection method [G. M. Shroff and H. B. Keller, SIAM J. Numer. Anal., 30 (1993), pp. 1099--1120] which identifies the (typically) low-dimensional slow dynamics of the (dissipative) model with a second reduction to the low-dimensional subspace of the decision variables. This results in the solution of a low-order unconstrained optimization problem. Optimal conditions are then computed in an efficient way using only low-dimensional numerical approximations of gradients and Hessians. The tubular reacto...

Computational studies of the transient behavior of horizontal MOVPE reactors

Dynamic models of the mass transfer in horizontal reactors used for metalorganic vapor phase epitaxy (MOVPE) of compound semiconductors have been developed and used to study the duration of transients on the substrate during the growth of heterostructures. Dispersion of reactants during gas switching can be detrimental to the abruptness of the interfaces. A solution to this problem is the operation of MOVPE reactors at very high flow rates and low pressures, but this leads to low conversions of the expensive precursors. Our simulations indicate that critical Peclet numbers can be identified, beyond which no significant reduction in the duration of transients on the substrate occurs when increasing the gas velocity. The substrate always reached a new steady state much faster than the entire reactor. Symmetry of filling and purging with respect to transients was also observed. Performance diagrams connecting optimal operating conditions with reactor geometry have been constructed. Time-dependent models of the MOVPE process can help identify optimal operating conditions and reactor shapes leading to abrupt interfaces with maximum precursor utilization.

Modeling Bacterial Population Growth from Stochastic Single-Cell Dynamics

ABSTRACT A few bacterial cells may be sufficient to produce a food-borne illness outbreak, provided that they are capable of adapting and proliferating on a food matrix. This is why any quantitative health risk assessment policy must incorporate methods to accurately predict the growth of bacterial populations from a small number of pathogens. In this aim, mathematical models have become a powerful tool. Unfortunately, at low cell concentrations, standard deterministic models fail to predict the fate of the population, essentially because the heterogeneity between individuals becomes relevant. In this work, a stochastic differential equation (SDE) model is proposed to describe variability within single-cell growth and division and to simulate population growth from a given initial number of individuals. We provide evidence of the model ability to explain the observed distributions of times to division, including the lag time produced by the adaptation to the environment, by comparing model predictions with experiments from the literature for Escherichia coli, Listeria innocua, and Salmonella enterica. The model is shown to accurately predict experimental growth population dynamics for both small and large microbial populations. The use of stochastic models for the estimation of parameters to successfully fit experimental data is a particularly challenging problem. For instance, if Monte Carlo methods are employed to model the required distributions of times to division, the parameter estimation problem can become numerically intractable. We overcame this limitation by converting the stochastic description to a partial differential equation (backward Kolmogorov) instead, which relates to the distribution of division times. Contrary to previous stochastic formulations based on random parameters, the present model is capable of explaining the variability observed in populations that result from the growth of a small number of initial cells as well as the lack of it compared to populations initiated by a larger number of individuals, where the random effects become negligible.