(404d) A Multiresolution Approach to Optimally Control the Dynamic Directed Self-Assembly of Nanostructures

Self-assembled nanoscale structures are the basis of various technological advancements such as molecular circuits and factories [1]. With significant progress in self-assembly of periodic nanostructures (such as monolayers [2]), the focus is now shifting towards non-periodic structures. Control of various interaction force fields (electrostatic, Van der Waals, etc.) [3] between the nanoparticles and external actuators can result in the formation of nanostructures with desired geometry. The aim is to design the nanoparticles and the external actuators such that the desired structure can be self-assembled rapidly with high reliability and avoiding any kinetic trapping that an ill-designed energy landscape might cause.

     Lakerveld et al. [4] showed that a master equation based dynamic model can be used to account for kinetic trapping explicitly. However, their model will quickly become intractable when designing larger systems. Therefore, the Adaptive Finite State Projection method [5] was introduced to deal with simulation of directed self-assembly in larger domains. The Adaptive Finite State Projection method identifies the important configurations and thus, simulates a manageable number of master equations.

     In this contribution, control of the self-assembly process is achieved using a combination of the multiresolution approach [1] and the Adaptive Finite State Projection method. The proposed method also restricts the physical domain, which enhances the Adaptive Finite State Projection method and mitigates the increasing size of the model. In the multiresolution approach, a desired nanostructure is developed in stages by dividing the physical domain into various partitions, and ensuring that these partitions correspond to ergodic subdomains. The physical domain is divided at the lowest resolution initially, i.e. two parts, in the first stage, which is then refined, i.e. four parts, eight parts and so on, in the subsequent stages. Every stage of this multiresolution approach aims to isolate a desired number of nanoparticles in each part of the domain, which is based on the final desired structure. Since the domain under consideration is too large to be simulated in detail, some of the state space is made inaccessible using repulsive external actuators and the remaining accessible part is simulated using the Adaptive Finite State Projection method. The external actuators are positioned and varied in such a way that the desired number of nanoparticles is obtained in each part of the divided domain, with a high probability in the minimum possible time. The time varying strengths of the external actuators are obtained by solving an optimization problem that minimizes the final time. The optimizer uses the Adaptive Finite State Projection innovatively to simulate the system and calculate the sensitivities for optimization.

     Once the isolation of nanoparticles in their respective parts of the domain is achieved, the particles are locked in those parts of the domain using external repulsive actuators and those parts are further refined in the next stage of the multiresolution approach. This process is repeated till the final desired structure is obtained. The proposed method is a novel mechanism to generate a nanostructure of desired geometry, which can be accurately simulated and effectively avoids kinetic traps during the process.

References

[1] N. Stephanopoulos, E. O. P. Solis, and G. Stephanopoulos, “Nanoscale process systems engineering: Toward molecular factories, synthetic cells, and adaptive devices," AIChE Journal, vol. 51, no. 7, pp. 1858-1869, 2005.

[2] J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology," Chemical Reviews, vol. 105, no. 4, pp. 1103-1170, 2005.

[3] K. J. M. Bishop, C. E. Wilmer, S. Soh, and B. A. Grzybowski, “Nanoscale forces and their uses in self-assembly," Small, vol. 5, no. 14, pp. 1600-1630, 2009.

[4] R. Lakerveld, G. Stephanopoulos, and P. I. Barton, “A master-equation approach to simulate kinetic traps during directed self-assembly," The Journal of Chemical Physics, vol. 136, no. 18, 2012.

[5] S. Ramaswamy, R. Lakerveld, G. Stephanopoulos, and P. I. Barton, “Approximating the solution to the master equation to simulate directed self-assembly of nanostructures," AIChE Annual Meeting, poster session, 2012.