Deterministic dynamic modeling of self-assembled nanostructures, directed by external fields, through a master equation approach, leads to a set of differential equations of such large size that even the most efficient solution algorithms are overwhelmed. Thus, model reduction is a key necessity. This paper presents a methodological approach and specific algorithms, which generate time-varying, reduced-order models for the description of directed self-assembly of nanoparticles by external fields. The approach is based on finite state projection and is adaptive; that is, it generates reduced-order models that vary over time. The algorithm uses event-detection concepts to determine automatically, during simulation, suitable time points at which the projection space and thus the structure of the reduced-order model change, in such a way that the computational load remains low while the maximum simulation error, resulting from model reduction, is lower than a prescribed upper bound. Such a model reduction technique aligns well with a control strategy that modifies the strengths and locations of the external charges that direct the self-assembly, in order for the self-assembling system to achieve the desired geometry, while avoiding any kinetic traps. The paper also presents a series of case studies, which illustrate how the proposed method can be used to simulate effectively the directed self-assembly of an appreciable number of nanoparticles, avoid kinetic traps, and reach the desired geometry. These case studies will also illustrate several properties of the proposed methodology.
ASJC Scopus subject areas
- Chemical Engineering(all)
- Industrial and Manufacturing Engineering