This research will develop conceptual design tools for the reliable fabrication of nanoscale structures with complex, non-periodic, and generally non-dense geometries. Self-assembly has been used previously to construct simple periodic nanostructures (thin films, 2-D templates on surfaces, and 3-D bulk materials structures) from nanoparticle building blocks such as colloidal particles, metal particles, and DNA. However, it is not presently possible to fabricate more complex, non-periodic and/or nondense structures with sufficient reliability. The goal here is to explore novel formulations, models and algorithms with the aim of developing a suite of conceptual design tools to address reliable fabrication of complex nanoscale structures.
The work will study the influence of external controls (such as nanoelectrodes, the system temperature, etc.) on the self-assembly behavior of specially functionalized nanoparticles, with the aim of developing optimal directed-assembly strategies that achieve high yields of the desired product. The self-assembly dynamics of nanoparticles (such as DNA tiles) will be modeled using master equations which consider the impact of shape, size, rotation, and both short- and long-range interactions among the particles and with external controls: Coulombic interactions, Van der Waals forces, hydrogen bonding of complementary DNA base pairs, and others. A multi-resolution, top-down approach will mitigate the combinatorics of the model to tractable levels, enabling deterministic dynamic optimization of the master equation to maximize the final probability of the desired configuration. For larger-scale problems, dynamic optimization based on sampling of the master equation will also be considered.
New algorithms will also be developed to determine the most reliable method for fabrication of a complex nanostructure from ?sub-assemblies.? These algorithms will determine the optimal way in which nanoparticles should be functionalized (such as with specific DNA sequences), as well as the step-by-step ?recipe? for the assembly of these nanoparticles into the final nanostructure. For any given nanostructure, the algorithms will also determine the optimal dynamic profiles for external controls, again to maximize the yield of the desired product. These results will be applicable to complex two-dimensional nanostructures and generalized for many types of nanoparticle building blocks.
Intellectual Merit
The conceptual design tools, formulations and methods that result from this effort will enable advances in the ability to design and reliably fabricate nanostructures for a wide range of applications. By making significant headway in the problem of reliable fabrication of complex, nonperiodic, and non-dense geometries, the focus of future nanoscale materials research can advance toward useful applications of specialized nanostructures and increased commercialization. The reliable fabrication of non-periodic nanostructures will open the door for new applications in a broad range of disciplines including nanoelectronic circuitry, molecular computing, artificial tissues, nanoscale chemical plants, high-sensitivity sensors, biodiagnostics (detection of proteins and DNA), plasmonic nanoparticle waveguides and other plasmonic devices, human tissue machine interfaces, medical devices, agricultural applications, and many others.
Broader Impact
The results of this work will be broadly disseminated through journal articles, conference presentations, publicly distributed software, and course curricula. The resulting models, algorithms and software will be made freely available to the scientific community. Personnel will be selected for this project from MIT?s diverse pool of graduate student researchers, which has successfully achieved high levels of minority and female enrollment. The PIs attract Ph.D. students from many ethnic backgrounds and a broad range of engineering disciplines, including chemical, mechanical, environmental, and biological engineering. MIT offers a diverse range of advanced-level courses on topics relevant for the research, enhancing the opportunities for multi-disciplinary research and education.
Self-assembly is the process of forming an ordered structure from initially disordered components that only interact locally, without external direction. At the molecular level, this process is a common technique for fabrication of nanostructures with periodic patterns. Self-assembled nanostructures usually demonstrate periodic patterns that only depend on the nature of their components and the environmental conditions under which the patterns are formed. However, several applications require fabrication of nanostructures with certain non-periodic geometries. The goal of this research is to answer whether this process can be directed by external actuators (e.g., nanoelectrodes, temperature, etc.) to fabricate nanostructures of desired geometry which are not necessarily periodic. In directed self-assembly, a number of charged nanoparticles (e.g., DNA tiles) are manipulated by external electrical fields (or temperature) to form a nanostructure of desired geometry. The directing electrical fields are generated and controlled by relatively small number of nanoelectrodes (compared to the number of particles) located at fixed locations on the substrate containing the particles. The dynamics of the particles are primarily governed by the interactions between them (self-assembly), and is modified to some extent by manipulation of the electrical potentials of these electrodes (external direction). The particles are initially distributed randomly on the substrate and are perturbed by random disturbances during the assembly process. Since the particle positions cannot be measured during the course of control, a feedback loop cannot be established and the electrodes are actuated only by open-loop controls. The control objective is to direct the particles towards formation of a desired pattern despite the uncertainty in their dynamics and initial positions. Under an optimal design, this control must maximize the probability of forming the desired pattern by the end of the assembly process, and maintain the formed structure under a static control afterward. Within the framework of these requirements, the specific objectives of this research has been: Development of mathematically tractable models to describe the dynamical behavior of the particles and the influence of actuations. Development of analytical or numerically tractable control design techniques. Development of numerically tractable simulation techniques to verify the control performance based on accurate dynamical models. This research introduced two different but closely related models to describe directed self-assembly: spatially continuous and spatially discrete models. The continuous model allows the particles to position continuously at any arbitrary point in a grid. This model is precise for larger particles of micrometer diameter and its continuous nature facilitates further analysis. The discrete model was developed for nanoscale particles, and it was shown how the results of the continuous model can be tailored to this discrete model only by minor modifications. Directed self-assembly along a straight line (1D grid) was extensively studied. This 1D special case possesses certain structure that allows for an exact and rigorous analysis and control design. The control design problem was formulated as a sequence of constrained optimization problems, and simplifying approximations and numerical techniques were proposed for resolving these optimization problems. Unlike the 1D case, design of a dynamic control for a 2D problem is not straightforward. This problem was formulated as an optimal control problem with infinite-dimensional dynamics represented by the Fokker-Planck partial differential equation. However, an exact solution to such an infinite-dimensional problem seems infeasible to obtain. An alternative methodology based on the concept of homotopy continuation was developed to reformulate the problem approximately as an optimal control problem with finite-dimensional dynamics. With a broader impact, this methodology is applicable to a wide class of nonlinear stochastic systems with multiple stable equilibria. In this proposed method, the state of the system is driven inside a potential canal from a unique initial stable equilibrium towards a certain final stable equilibrium with a minimized probability of escaping the canal. The potential canal as introduced in this work is a dynamic extension of the existing concept of potential well and is a continuous trajectory of minimum energy that connects an initial stable equilibrium to a desired stable equilibrium out of the multiple stable equilibria created by a static control. The performance of a designed control must be evaluated by numerical simulations; however, the scope of this method is limited by complexity of the equations that govern the dynamics of the interacting particles. A natural alternative is to utilize Monte Carlo methods by simulating a large number of state trajectories and extract a quantity of interest by averaging over these trajectories. A second alternative is to reduce the dimension of these dynamical equations. Using a proposed Adaptive Finite State Projection method, the original equations were reduced in dimension, so that they can be numerically solved with reasonable computational cost.
