7746
Ind. Eng. Chem. Res. 2010, 49, 7746–7757
Controlled Formation of Nanostructures with Desired Geometries. 2. Robust Dynamic Paths Earl O. P. Solis, Paul I. Barton, and George Stephanopoulos* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
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. 1. Introduction: The Dynamic Problem In part 1 of this series, we addressed the static problem associated with the controlled formation of nanostructures with desired geometries. We were able to guarantee a statistically robust desired structure by solving the energy-gap maximization problem (EMP).1 We showed that we could reduce the phase space combinatorics and include only the neighboring competing states, while still guaranteeing a robust desired final structure. Given that we have introduced nonergodicity to the system energy landscape, i.e., we have created a rough energy landscape that traps the system in subsets of phase space (called components),2 arriving at the desired system configuration from any initial configuration can only occur if the system begins in the desired configuration’s component, a requirement that is impossible to realize by existing technologies. Starting from an arbitrary distribution of particles in the complete phase space, the probability of reaching the desired nanostructure, under the influence of controls determined by the solution of the static problem in part 1, is unacceptably low and variable, since the system can be trapped in different metastable states, which are determined by the dynamic path of the self-assembly process. Following the definitions we introduced in part 1, the term component will denote a set of configurations that are accessible from each other, i.e., a subset of the set of configurations (states) in the complete phase space composed of kinetically accessible states from any state that is also a member of the component. In this paper, we will develop a methodology that overcomes the above restriction and allows the dynamically self-assembled particles to reach the desired final geometry from any initial configuration with prespecified high probability, i.e., generate robust dynamic self-assembly paths. * To whom correspondence should be addressed. E-mail: geosteph@ mit.edu.
As in part 1, the model system we use to demonstrate the proposed robust strategies for controlled self-assembly is an isomorph of the lattice Ising model.3 We assume that the system particles are radially symmetric and simply translate within a finite system volume. We also assume that all the particles are the same. Analyzing only the positional probability of a particular configuration, the potential energy function is given by Nd
V
E(z) ) Eext(z) + Eint(z) )
∑ ∑zH i
i,ksk
+
i)1 k)1
∑zJ
i i,jzj
)
i
