There is a long history associated with the mathematical analysis of chemical reaction dynamics to generate reduced-order models, particularly in homogeneous reaction systems. Combustion reaction dynamics research, for example, has produced such studies as Maas and Pope (1992) and Lam and Goussis (1994); the process systems community has contributed significantly as well (e.g., Vlachos, 1996; Vora and Daoutidis, 2001). By comparison, heterogeneous reacting systems have received less attention, especially in the realm of thin-film processing.

Effective modeling of heterogeneous reaction dynamics is crucial to Atomic Layer Deposition (ALD) high-throughput system process design and optimization. In ALD, thin films are deposited in discrete (usually sub-monolayer) increments by exposing the growth surface to the gas-phase reactive precursors in an alternating manner. Unlike chemical vapor deposition, ALD does not possess steady state growth modes; instead, continuous ALD operation consists of limit-cycle behavior.

Generating accurate experimental data useful for studying intrinsic ALD reaction rates is challenging because of the dynamic nature of the process, the wide time-scale range of the dynamic processes, and the small surface concentrations of some of the reaction intermediate species. Therefore, researchers have turned to quantum chemical computations to model and compare ALD reaction mechanisms (Elliott, 2012). Because these methods primarily produce static information regarding reaction energetics and transition state configurations, the transitions between surface states can only be described in terms of equilibrium relations. However, conventional transition-state theory does provide rate expressions for the activated surface reactions, and rate equations for precursor species adsorption and desorption processes likewise can be formulated.

Given the modeling situation described, we should expect the overall ALD reaction network (RN) model to consist of a differential-algebraic equation (DAE) system; a key objective of our work in this area has been to develop rational methods to formulate ALD reaction kinetics models in the form of a well-posed DAE system. By writing all reactions in terms of their net-forward rates, including equilibrium processes by adding a fictitious time constant $\epsilon << 1$ s, the pure differential-equation system then can be factored to decouple the reaction terms (Remmers, et. al, 2015; Vora and Daoutidis, 2001). Success of this factorization procedure indicates that the outer solution to this singular perturbation problem retains the dynamics required to accurately model the deposition process. A number of reaction-independent modes also are normally produced by this process, reflecting the conserved quantities of the process and elimination of the redundant dynamic modes. Computing the span of the species space associated with the slow and conserved modes while ignoring minor surface species can further reduce the dynamic dimension of the deposition system by identifying combinations of finite-rate processes that approximate new equilibrium relationships.

The stoichiometry array that premultiplies the reaction-rate array of the model described is a adjacency matrix connecting reaction rates to the time-rate of change of the reaction species, and so can be used to create a (bipartite) species-reaction (SR) graph. The SR graph has been used previously to analyze RN for the potential of multiplicity (Craciun and Feinberg, 2006). Here we will use SR graphs to provide additional insight into our factorization procedure and to diagnose structural problems within thin-film surface RN models.

References

Craciun, G. and M. Feinberg, Multiple equilibria in complex chemical reaction networks II: The species-reaction graph, SIAM J. Appl. Math. 66 1320-1338 (2006).

Elliott, S. D., Atomic-scale simulation of ALD chemistry, Semicond. Sci. Technol. 27 074008 (2012).

Lam, S. H. and D. A. Goussis, The CSP method for simplifying kinetics, Int. J. Chem. Kinetics 26 461-486 (1994).

Maas, U. and S. B. Pope, Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space, Combustion and Flame 88 239-264 (1992).

Remmers, E. M., C. D. Travis, and R. A. Adomaitis, Reaction factorization for the dynamic analysis of atomic layer deposition kinetics, Chemical Engineering Science 127 374-391 (2015).

Vlachos, D. G., Reduction of detailed kinetic mechanisms for ignition and extinction of premixed hydrogen/air flames, Chem. Engng Sci. 51 3979-3993 (1996).

Vora, N. and P. Daoutidis, Nonlinear model reduction of chemical reaction systems, AIChE J. 47 2320-2332 (2001).

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