Spatially distributed multiscale systems have been recently developed to describe transport-reaction processes for which phenomena occur across length scales that differ by several orders of magnitude. These descriptions can range from the smaller regimes (microscale, quantum and atomistic) to the intermediate (mesoscale) and larger (macroscale/continuum) ones. The level of detail required to accurately describe the dynamic behavior of the system at smaller length scales cannot be provided by solely using a description from larger length-scales. On the other hand, describing larger length-scale behavior using descriptions from smaller length-scale models would be infeasible due to the required computational overhead of this procedure. Motivated by the above, multiscale models, which combine larger and smaller-length scale models that are intimately connected, have been developed.

Such multiscale models could prove very useful in optimization, as continuous pressure on profit margins has resulted in the desire to improve efficiency in order to reduce costs. Multiscale optimization of such processes, however, is a problem with three levels of hierarchy in computation. In the lower hierarchy, potentially computationally intensive simulations may need to be performed to model the properties of interest at both smaller and larger length-scales. If bi-directional information (conditions at a larger length-scale affect the conditions at a smaller length-scale and vice-versa) is required, both these smaller and larger length-scale simulations must be performed iteratively at each process condition in order to achieve convergence of the multiscale description, which is the intermediate hierarchy. Finally, gradient information might not be available in closed form, necessitating the use of black-box optimization algorithms. Black-box optimization can require several function evaluations, which in this case is at the top of the hierarchy, and therefore all of these evaluations must include reevaluation of the process behavior for all length scales involved. With the need to perform numerous simulations, solving a multiscale optimization problem could quickly become intractable. Many different methods have been proposed to circumvent this computational intractability. One promising idea is to obtain the necessary information from linear interpolations whenever possible instead of using costly simulations. This is the basic idea behind in situ adaptive tabulation (ISAT). Initially used in solving combustion chemistry problems [1], ISAT has been subsequently extended to stochastic systems and combined with black-box optimization for maximizing uniformity and minimizing surface roughness in a gallium nitride thin-film [2]. Additionally, if lattice-based atomistic simulations are used, lattice reconstruction techniques based on the low-order properties of interest can be employed [3].

The implementation and further refinement of these methods is investigated by modeling the deposition of a thin-film consisting of alternating gallium arsenide and aluminum arsenide layers (GaAs/AlAs.) This example uses macroscale “inputs” from the reactor description [2,4] to determine properties such as temperature, flow rates, and concentrations, and mesoscale kinetic Monte Carlo (kMC) simulations to measure the previously characterized interfacial properties of the film. The objectives of this problem are to minimize the interfacial step-densities between GaAs and AlAs layers, while also minimizing the temperature and the time spent in-between depositing species (termed annealing time,) as well the macroscopic objective of reducing spatial non-uniformity. This problem only involves adsorption and therefore only requires uni-directional flow of information from the macroscale level to the mesoscale level. Based on this model, we explore the methodology needed to accelerate the computational process by using techniques such as ISAT. Using this methodology, a multiscale model for a GaAs/AlAs thin-film deposition process was developed for optimization of the interfacial properties, and used within an efficient optimization framework to identify optimal-time varying process conditions for the process.

1. S.B. Pope, “Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation,” Combust. Theory Modelling, 1, 41-63, 1997.

2. A. Varshney and A. Armaou, “Multiscale optimization using hybrid PDE/kMC process systems with application to thin film growth,” Chem. Eng. Sci., 60, 6780-6794, 2005.

3. A. Varshney and A. Armaou, “Identification of macroscopic variables for low-order modeling of thin-film growth,” Ind. Eng. Chem. Res., 45, 8290-8298, 2006.

4. A. Varshney and A. Armaou, “Reduced order modeling and dynamic optimization of multiscale PDE/kMC process systems,” Comp. & Chem. Eng., in press, 2008.