Controlled formation of nanostructures with desired geometries. 2. Robust dynamic paths

Part 2 of this series addresses the question of how to manipulate in time the positions and intensities of external controls so that a set of self-assembling nanoscale particles can always reach the nanostructure of desired geometry, starting from any random and unknown spatial distribution of the particles. It complements part 1 in which we examined how to position external controls and compute their intensities so that we can ensure that the final nanostructure with the desired geometry corresponds to a local potential energy minimum surrounded by sufficiently high energy barriers to ensure that the nanostructure is statistically robust, i.e., it remains at the desired geometry with an acceptably high probability. The proposed approach for the generation of robust dynamic self-assembly paths is based on a progressive reduction of the system phase space into subsets with progressively smaller numbers of locally allowable configurational states. In other words, it is based on a judicious progressive transition from ergodic to nonergodic subsystems. The subsets of allowable configurations in phase space are modeled by a wavelet-based spatial multiresolution view of the desired structure (in terms of the number of particles). This approach produces a prescription of the optimal control problem where the dynamic self-assembly of particles into the desired nanostructure is governed by the dynamic master equation of statistical mechanics. A genetic algorithm is used to solve the associated optimization problems at each time period and locate the position of the external controls in the physical domain, as well as their intensities over time. The approaches and methods are illustrated with 1- and 2-dimensional lattice example systems.