A master-equation approach to simulate kinetic traps during directed self-assembly

Abstract

Robust directed self-assembly of non-periodic nanoscale structures is a key process that would enable various technological breakthroughs. The dynamic evolution of directed self-assemblies towards structures with desired geometries is governed by the rugged potential energy surface of nanoscale systems, potentially leading the system to kinetic traps. To study such phenomena and to set the framework for the directed self-assembly of nanoparticles towards structures with desired geometries, the development of a dynamic model involving a master equation to simulate the directed self-assembly process is presented. The model describes the probability of each possible configuration of a fixed number of nanoparticles on a domain, including parametric sensitivities that can be used for optimization, as a function of time during self-assembly. An algorithm is presented that solves large-scale instances of the model with linear computational complexity. Case studies illustrate the influence of several degrees of freedom on directed self-assembly. A design approach that systematically decomposes the ergodicity of the system to direct self-assembly of a targeted configuration with high probability is illustrated. The prospects for extending such an approach to larger systems using coarse graining techniques are also discussed.

Controlled Formation of Nanostructures with Desired Geometries. Part 4. Multiresolution Optimal Control in Dynamically Directed Self-Assembly of Nanoparticles

“…The probability of observing the nanoparticles in a particular configuration as a function of time is given by a master equation: 1,5 …”

Potential Canals for Control of Nonlinear Stochastic Systems in the Absence of State Measurements

“…The control scheme in this paper was primarily developed for use in directed self-assembly of nanoparticles [11]- [19]. Self-assembly is the process of forming an ordered structure from initially disordered components that only interact locally, without external direction.…”

“…In many applications, however, nanostructures with specific nonperiodic geometries are required. To address the need for fabrication of nanostructures with specific geometries, researchers proposed to direct the self-assembly process towards the desired geometry using external controls [11]- [19].…”

Control of self-assembly in micro- and nano-scale systems

“…Local manipulation of fields (e.g., temperature, concentration, pH, electric field, and magnetic field) has been shown in simulations to drive the system toward a particular desired configuration with a high probability [52,53,55]. Methods for qualitative and quantitative shaping of the energy landscape through manipulation of these fields were reviewed in Section 3.2.…”

“…This no-measurements situation naturally gives rise to stochastic models for describing the system dynamics such as (1) and (3). In [55], a Master equation model was used in conjunction with the aforementioned methods of [52,53]. It is shown that open-loop control schemes can drive the self-assembly of the system (in simulations) to a targeted configuration with high probability.…”