Controlled formation of nanostructures with desired geometries: Part 3. Dynamic modeling and simulation of directed self-assembly of nanoparticles through adaptive finite state projection

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.